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Properties

Label 12-126e6-1.1-c7e6-0-3
Degree 1212
Conductor 4.002×10124.002\times 10^{12}
Sign 11
Analytic cond. 3.71847×1093.71847\times 10^{9}
Root an. cond. 6.273796.27379
Motivic weight 77
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 24·2-s + 192·4-s − 110·5-s + 635·7-s + 1.02e3·8-s + 2.64e3·10-s + 548·11-s + 1.98e4·13-s − 1.52e4·14-s − 3.68e4·16-s + 2.09e4·17-s + 2.83e4·19-s − 2.11e4·20-s − 1.31e4·22-s + 3.27e4·23-s + 1.34e5·25-s − 4.77e5·26-s + 1.21e5·28-s − 2.34e5·29-s − 1.47e5·31-s + 2.94e5·32-s − 5.03e5·34-s − 6.98e4·35-s − 3.67e5·37-s − 6.81e5·38-s − 1.12e5·40-s + 2.87e6·41-s + ⋯
L(s)  = 1  − 2.12·2-s + 3/2·4-s − 0.393·5-s + 0.699·7-s + 0.707·8-s + 0.834·10-s + 0.124·11-s + 2.51·13-s − 1.48·14-s − 9/4·16-s + 1.03·17-s + 0.949·19-s − 0.590·20-s − 0.263·22-s + 0.560·23-s + 1.72·25-s − 5.32·26-s + 1.04·28-s − 1.78·29-s − 0.891·31-s + 1.59·32-s − 2.19·34-s − 0.275·35-s − 1.19·37-s − 2.01·38-s − 0.278·40-s + 6.51·41-s + ⋯

Functional equation

Λ(s)=((26⋅312⋅76)s/2 ΓC(s)6 L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}
Λ(s)=((26⋅312⋅76)s/2 ΓC(s+7/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 26⋅312⋅762^{6} \cdot 3^{12} \cdot 7^{6}
Sign: 11
Analytic conductor: 3.71847×1093.71847\times 10^{9}
Root analytic conductor: 6.273796.27379
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 26⋅312⋅76, ( :[7/2]6), 1)(12,\ 2^{6} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [7/2]^{6} ),\ 1 )

Particular Values

L(4)L(4) ≈\approx 6.6506183366.650618336
L(12)L(\frac12) ≈\approx 6.6506183366.650618336
L(92)L(\frac{9}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 (1+p3T+p6T2)3 ( 1 + p^{3} T + p^{6} T^{2} )^{3}
3 1 1
7 1−635T−72748pT2+667945p4T3−72748p8T4−635p14T5+p21T6 1 - 635 T - 72748 p T^{2} + 667945 p^{4} T^{3} - 72748 p^{8} T^{4} - 635 p^{14} T^{5} + p^{21} T^{6}
good5 1+22pT−24502pT2−6966776pT3+194927678p2T4+67954540106p2T5+3210602234854p2T6+67954540106p9T7+194927678p16T8−6966776p22T9−24502p29T10+22p36T11+p42T12 1 + 22 p T - 24502 p T^{2} - 6966776 p T^{3} + 194927678 p^{2} T^{4} + 67954540106 p^{2} T^{5} + 3210602234854 p^{2} T^{6} + 67954540106 p^{9} T^{7} + 194927678 p^{16} T^{8} - 6966776 p^{22} T^{9} - 24502 p^{29} T^{10} + 22 p^{36} T^{11} + p^{42} T^{12}
11 1−548T−4276622T2−160652942204T3−24911416435726T4+22869026595445016pT5+ 1 - 548 T - 4276622 T^{2} - 160652942204 T^{3} - 24911416435726 T^{4} + 22869026595445016 p T^{5} + 16 ⁣⋯ ⁣2616\!\cdots\!26p2T6+22869026595445016p8T7−24911416435726p14T8−160652942204p21T9−4276622p28T10−548p35T11+p42T12 p^{2} T^{6} + 22869026595445016 p^{8} T^{7} - 24911416435726 p^{14} T^{8} - 160652942204 p^{21} T^{9} - 4276622 p^{28} T^{10} - 548 p^{35} T^{11} + p^{42} T^{12}
13 (1−9949T+49691323T2+136109955742T3+49691323p7T4−9949p14T5+p21T6)2 ( 1 - 9949 T + 49691323 T^{2} + 136109955742 T^{3} + 49691323 p^{7} T^{4} - 9949 p^{14} T^{5} + p^{21} T^{6} )^{2}
17 1−20972T−705540803T2+6376043535148T3+523135254353192486T4− 1 - 20972 T - 705540803 T^{2} + 6376043535148 T^{3} + 523135254353192486 T^{4} - 12 ⁣⋯ ⁣1612\!\cdots\!16T5− T^{5} - 24 ⁣⋯ ⁣0324\!\cdots\!03T6− T^{6} - 12 ⁣⋯ ⁣1612\!\cdots\!16p7T7+523135254353192486p14T8+6376043535148p21T9−705540803p28T10−20972p35T11+p42T12 p^{7} T^{7} + 523135254353192486 p^{14} T^{8} + 6376043535148 p^{21} T^{9} - 705540803 p^{28} T^{10} - 20972 p^{35} T^{11} + p^{42} T^{12}
19 1−28383T−1114469436T2+54482095066865T3+423775802745621540T4− 1 - 28383 T - 1114469436 T^{2} + 54482095066865 T^{3} + 423775802745621540 T^{4} - 15 ⁣⋯ ⁣9715\!\cdots\!97pT5+ p T^{5} + 12 ⁣⋯ ⁣5412\!\cdots\!54p2T6− p^{2} T^{6} - 15 ⁣⋯ ⁣9715\!\cdots\!97p8T7+423775802745621540p14T8+54482095066865p21T9−1114469436p28T10−28383p35T11+p42T12 p^{8} T^{7} + 423775802745621540 p^{14} T^{8} + 54482095066865 p^{21} T^{9} - 1114469436 p^{28} T^{10} - 28383 p^{35} T^{11} + p^{42} T^{12}
23 1−32732T−4257974405T2+384004545798916T3+1117012122760959974T4− 1 - 32732 T - 4257974405 T^{2} + 384004545798916 T^{3} + 1117012122760959974 T^{4} - 67 ⁣⋯ ⁣4467\!\cdots\!44T5+ T^{5} + 47 ⁣⋯ ⁣8747\!\cdots\!87T6− T^{6} - 67 ⁣⋯ ⁣4467\!\cdots\!44p7T7+1117012122760959974p14T8+384004545798916p21T9−4257974405p28T10−32732p35T11+p42T12 p^{7} T^{7} + 1117012122760959974 p^{14} T^{8} + 384004545798916 p^{21} T^{9} - 4257974405 p^{28} T^{10} - 32732 p^{35} T^{11} + p^{42} T^{12}
29 (1+117088T+32964581490T2+4122277361969500T3+32964581490p7T4+117088p14T5+p21T6)2 ( 1 + 117088 T + 32964581490 T^{2} + 4122277361969500 T^{3} + 32964581490 p^{7} T^{4} + 117088 p^{14} T^{5} + p^{21} T^{6} )^{2}
31 1+147865T+804539397T2+5946198810653250T3+73494989034467037151T4− 1 + 147865 T + 804539397 T^{2} + 5946198810653250 T^{3} + 73494989034467037151 T^{4} - 89 ⁣⋯ ⁣9589\!\cdots\!95T5+ T^{5} + 14 ⁣⋯ ⁣3814\!\cdots\!38T6− T^{6} - 89 ⁣⋯ ⁣9589\!\cdots\!95p7T7+73494989034467037151p14T8+5946198810653250p21T9+804539397p28T10+147865p35T11+p42T12 p^{7} T^{7} + 73494989034467037151 p^{14} T^{8} + 5946198810653250 p^{21} T^{9} + 804539397 p^{28} T^{10} + 147865 p^{35} T^{11} + p^{42} T^{12}
37 1+367503T−25474888698T2+9103014313145033T3+ 1 + 367503 T - 25474888698 T^{2} + 9103014313145033 T^{3} + 13 ⁣⋯ ⁣3613\!\cdots\!36T4− T^{4} - 38 ⁣⋯ ⁣6138\!\cdots\!61T5− T^{5} - 13 ⁣⋯ ⁣0813\!\cdots\!08T6− T^{6} - 38 ⁣⋯ ⁣6138\!\cdots\!61p7T7+ p^{7} T^{7} + 13 ⁣⋯ ⁣3613\!\cdots\!36p14T8+9103014313145033p21T9−25474888698p28T10+367503p35T11+p42T12 p^{14} T^{8} + 9103014313145033 p^{21} T^{9} - 25474888698 p^{28} T^{10} + 367503 p^{35} T^{11} + p^{42} T^{12}
41 (1−1437954T+1203779247675T2−639058142642674020T3+1203779247675p7T4−1437954p14T5+p21T6)2 ( 1 - 1437954 T + 1203779247675 T^{2} - 639058142642674020 T^{3} + 1203779247675 p^{7} T^{4} - 1437954 p^{14} T^{5} + p^{21} T^{6} )^{2}
43 (1−749397T+509340737649T2−170374599838662482T3+509340737649p7T4−749397p14T5+p21T6)2 ( 1 - 749397 T + 509340737649 T^{2} - 170374599838662482 T^{3} + 509340737649 p^{7} T^{4} - 749397 p^{14} T^{5} + p^{21} T^{6} )^{2}
47 1+741486T−864917943045T2−363526634183852922T3+ 1 + 741486 T - 864917943045 T^{2} - 363526634183852922 T^{3} + 76 ⁣⋯ ⁣3476\!\cdots\!34T4+ T^{4} + 12 ⁣⋯ ⁣4212\!\cdots\!42T5− T^{5} - 39 ⁣⋯ ⁣5739\!\cdots\!57T6+ T^{6} + 12 ⁣⋯ ⁣4212\!\cdots\!42p7T7+ p^{7} T^{7} + 76 ⁣⋯ ⁣3476\!\cdots\!34p14T8−363526634183852922p21T9−864917943045p28T10+741486p35T11+p42T12 p^{14} T^{8} - 363526634183852922 p^{21} T^{9} - 864917943045 p^{28} T^{10} + 741486 p^{35} T^{11} + p^{42} T^{12}
53 1+1032432T−2372684155170T2−1282239454997266536T3+ 1 + 1032432 T - 2372684155170 T^{2} - 1282239454997266536 T^{3} + 52 ⁣⋯ ⁣5452\!\cdots\!54T4+ T^{4} + 13 ⁣⋯ ⁣8413\!\cdots\!84T5− T^{5} - 62 ⁣⋯ ⁣9862\!\cdots\!98T6+ T^{6} + 13 ⁣⋯ ⁣8413\!\cdots\!84p7T7+ p^{7} T^{7} + 52 ⁣⋯ ⁣5452\!\cdots\!54p14T8−1282239454997266536p21T9−2372684155170p28T10+1032432p35T11+p42T12 p^{14} T^{8} - 1282239454997266536 p^{21} T^{9} - 2372684155170 p^{28} T^{10} + 1032432 p^{35} T^{11} + p^{42} T^{12}
59 1+2389238T−54827765486T2−423685096884345220T3+ 1 + 2389238 T - 54827765486 T^{2} - 423685096884345220 T^{3} + 19 ⁣⋯ ⁣9019\!\cdots\!90T4− T^{4} - 12 ⁣⋯ ⁣0212\!\cdots\!02T5− T^{5} - 36 ⁣⋯ ⁣6636\!\cdots\!66T6− T^{6} - 12 ⁣⋯ ⁣0212\!\cdots\!02p7T7+ p^{7} T^{7} + 19 ⁣⋯ ⁣9019\!\cdots\!90p14T8−423685096884345220p21T9−54827765486p28T10+2389238p35T11+p42T12 p^{14} T^{8} - 423685096884345220 p^{21} T^{9} - 54827765486 p^{28} T^{10} + 2389238 p^{35} T^{11} + p^{42} T^{12}
61 1+1238746T−3495698274519T2−5946787860873446010T3+ 1 + 1238746 T - 3495698274519 T^{2} - 5946787860873446010 T^{3} + 19 ⁣⋯ ⁣3019\!\cdots\!30T4+ T^{4} + 36 ⁣⋯ ⁣6636\!\cdots\!66T5+ T^{5} + 44 ⁣⋯ ⁣4144\!\cdots\!41T6+ T^{6} + 36 ⁣⋯ ⁣6636\!\cdots\!66p7T7+ p^{7} T^{7} + 19 ⁣⋯ ⁣3019\!\cdots\!30p14T8−5946787860873446010p21T9−3495698274519p28T10+1238746p35T11+p42T12 p^{14} T^{8} - 5946787860873446010 p^{21} T^{9} - 3495698274519 p^{28} T^{10} + 1238746 p^{35} T^{11} + p^{42} T^{12}
67 1−2462497T−13268883047308T2+11883615447576942903T3+ 1 - 2462497 T - 13268883047308 T^{2} + 11883615447576942903 T^{3} + 18 ⁣⋯ ⁣7618\!\cdots\!76T4− T^{4} - 88 ⁣⋯ ⁣0188\!\cdots\!01T5− T^{5} - 11 ⁣⋯ ⁣9811\!\cdots\!98T6− T^{6} - 88 ⁣⋯ ⁣0188\!\cdots\!01p7T7+ p^{7} T^{7} + 18 ⁣⋯ ⁣7618\!\cdots\!76p14T8+11883615447576942903p21T9−13268883047308p28T10−2462497p35T11+p42T12 p^{14} T^{8} + 11883615447576942903 p^{21} T^{9} - 13268883047308 p^{28} T^{10} - 2462497 p^{35} T^{11} + p^{42} T^{12}
71 (1−636262T+18637597322721T2−19723903310040373948T3+18637597322721p7T4−636262p14T5+p21T6)2 ( 1 - 636262 T + 18637597322721 T^{2} - 19723903310040373948 T^{3} + 18637597322721 p^{7} T^{4} - 636262 p^{14} T^{5} + p^{21} T^{6} )^{2}
73 1+4609961T−504249505402T2+439056397955388279T3+ 1 + 4609961 T - 504249505402 T^{2} + 439056397955388279 T^{3} + 28 ⁣⋯ ⁣1628\!\cdots\!16T4− T^{4} - 56 ⁣⋯ ⁣0756\!\cdots\!07T5− T^{5} - 26 ⁣⋯ ⁣1226\!\cdots\!12T6− T^{6} - 56 ⁣⋯ ⁣0756\!\cdots\!07p7T7+ p^{7} T^{7} + 28 ⁣⋯ ⁣1628\!\cdots\!16p14T8+439056397955388279p21T9−504249505402p28T10+4609961p35T11+p42T12 p^{14} T^{8} + 439056397955388279 p^{21} T^{9} - 504249505402 p^{28} T^{10} + 4609961 p^{35} T^{11} + p^{42} T^{12}
79 1−6152849T−9996162298563T2+91000866673663082982T3+ 1 - 6152849 T - 9996162298563 T^{2} + 91000866673663082982 T^{3} + 18 ⁣⋯ ⁣7918\!\cdots\!79T4− T^{4} - 18 ⁣⋯ ⁣9718\!\cdots\!97T5− T^{5} - 66 ⁣⋯ ⁣0666\!\cdots\!06T6− T^{6} - 18 ⁣⋯ ⁣9718\!\cdots\!97p7T7+ p^{7} T^{7} + 18 ⁣⋯ ⁣7918\!\cdots\!79p14T8+91000866673663082982p21T9−9996162298563p28T10−6152849p35T11+p42T12 p^{14} T^{8} + 91000866673663082982 p^{21} T^{9} - 9996162298563 p^{28} T^{10} - 6152849 p^{35} T^{11} + p^{42} T^{12}
83 (1−12029884T+1146704654226pT2− ( 1 - 12029884 T + 1146704654226 p T^{2} - 54 ⁣⋯ ⁣3854\!\cdots\!38T3+1146704654226p8T4−12029884p14T5+p21T6)2 T^{3} + 1146704654226 p^{8} T^{4} - 12029884 p^{14} T^{5} + p^{21} T^{6} )^{2}
89 1−10646976T−54532854429783T2+ 1 - 10646976 T - 54532854429783 T^{2} + 16 ⁣⋯ ⁣6816\!\cdots\!68T3+ T^{3} + 11 ⁣⋯ ⁣4611\!\cdots\!46pT4− p T^{4} - 26 ⁣⋯ ⁣2826\!\cdots\!28T5− T^{5} - 34 ⁣⋯ ⁣1134\!\cdots\!11T6− T^{6} - 26 ⁣⋯ ⁣2826\!\cdots\!28p7T7+ p^{7} T^{7} + 11 ⁣⋯ ⁣4611\!\cdots\!46p15T8+ p^{15} T^{8} + 16 ⁣⋯ ⁣6816\!\cdots\!68p21T9−54532854429783p28T10−10646976p35T11+p42T12 p^{21} T^{9} - 54532854429783 p^{28} T^{10} - 10646976 p^{35} T^{11} + p^{42} T^{12}
97 (1−2281472T+178988504234792T2− ( 1 - 2281472 T + 178988504234792 T^{2} - 22 ⁣⋯ ⁣4622\!\cdots\!46T3+178988504234792p7T4−2281472p14T5+p21T6)2 T^{3} + 178988504234792 p^{7} T^{4} - 2281472 p^{14} T^{5} + p^{21} T^{6} )^{2}
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   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−6.01973626278208092471213158021, −5.83166757748642801540207606544, −5.58211952386985561247781427824, −5.57218265079979756354541860092, −5.22935485224327541944287031491, −5.05739046439857781105933604082, −4.62631315490536333618286752295, −4.31238868873235493760234174345, −4.27723869605522784613479980349, −4.20767994870684937214676651382, −3.70247488275758409797491554616, −3.52893136519863071412030627101, −3.42456934179565188840308524022, −3.16796976258846330522294123548, −2.53960548936687602932248939393, −2.49664301078676126355645984811, −2.44514059241400705087522557982, −1.66757975580888226144647376085, −1.56925752569469154549875158329, −1.44712759021640165564308646118, −1.17353125681665586864323362846, −0.74733985993735758465978392881, −0.66595292770718486017135440211, −0.66449043974912199091388707117, −0.39601269354382914309386226796, 0.39601269354382914309386226796, 0.66449043974912199091388707117, 0.66595292770718486017135440211, 0.74733985993735758465978392881, 1.17353125681665586864323362846, 1.44712759021640165564308646118, 1.56925752569469154549875158329, 1.66757975580888226144647376085, 2.44514059241400705087522557982, 2.49664301078676126355645984811, 2.53960548936687602932248939393, 3.16796976258846330522294123548, 3.42456934179565188840308524022, 3.52893136519863071412030627101, 3.70247488275758409797491554616, 4.20767994870684937214676651382, 4.27723869605522784613479980349, 4.31238868873235493760234174345, 4.62631315490536333618286752295, 5.05739046439857781105933604082, 5.22935485224327541944287031491, 5.57218265079979756354541860092, 5.58211952386985561247781427824, 5.83166757748642801540207606544, 6.01973626278208092471213158021

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.