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Properties

Label 24-338e12-1.1-c3e12-0-3
Degree 2424
Conductor 2.223×10302.223\times 10^{30}
Sign 11
Analytic cond. 3.95724×10153.95724\times 10^{15}
Root an. cond. 4.465714.46571
Motivic weight 33
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  + 24·3-s + 12·4-s + 360·9-s + 288·12-s + 48·16-s − 180·17-s + 38·23-s + 872·25-s + 4.37e3·27-s + 202·29-s + 4.32e3·36-s − 2.41e3·43-s + 1.15e3·48-s − 1.21e3·49-s − 4.32e3·51-s − 4.38e3·53-s + 2.21e3·61-s − 128·64-s − 2.16e3·68-s + 912·69-s + 2.09e4·75-s − 7.84e3·79-s + 4.49e4·81-s + 4.84e3·87-s + 456·92-s + 1.04e4·100-s + 2.55e3·101-s + ⋯
L(s)  = 1  + 4.61·3-s + 3/2·4-s + 40/3·9-s + 6.92·12-s + 3/4·16-s − 2.56·17-s + 0.344·23-s + 6.97·25-s + 31.1·27-s + 1.29·29-s + 20·36-s − 8.54·43-s + 3.46·48-s − 3.53·49-s − 11.8·51-s − 11.3·53-s + 4.65·61-s − 1/4·64-s − 3.85·68-s + 1.59·69-s + 32.2·75-s − 11.1·79-s + 61.6·81-s + 5.97·87-s + 0.516·92-s + 10.4·100-s + 2.51·101-s + ⋯

Functional equation

Λ(s)=((212⋅1324)s/2 ΓC(s)12 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((212⋅1324)s/2 ΓC(s+3/2)12 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2424
Conductor: 212⋅13242^{12} \cdot 13^{24}
Sign: 11
Analytic conductor: 3.95724×10153.95724\times 10^{15}
Root analytic conductor: 4.465714.46571
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (24, 212⋅1324, ( :[3/2]12), 1)(24,\ 2^{12} \cdot 13^{24} ,\ ( \ : [3/2]^{12} ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 8.5199387828.519938782
L(12)L(\frac12) ≈\approx 8.5199387828.519938782
L(52)L(\frac{5}{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−p2T2+p4T4)3 ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{3}
13 1 1
good3 (1−4pT+4p2T2−26T3+376pT4−272p3T5+22687T6−272p6T7+376p7T8−26p9T9+4p14T10−4p16T11+p18T12)2 ( 1 - 4 p T + 4 p^{2} T^{2} - 26 T^{3} + 376 p T^{4} - 272 p^{3} T^{5} + 22687 T^{6} - 272 p^{6} T^{7} + 376 p^{7} T^{8} - 26 p^{9} T^{9} + 4 p^{14} T^{10} - 4 p^{16} T^{11} + p^{18} T^{12} )^{2}
5 (1−436T2+103056T4−15453889T6+103056p6T8−436p12T10+p18T12)2 ( 1 - 436 T^{2} + 103056 T^{4} - 15453889 T^{6} + 103056 p^{6} T^{8} - 436 p^{12} T^{10} + p^{18} T^{12} )^{2}
7 1+1213T2+846032T4+44311205pT6+45548573507T8−21505406726124T10−12859398267661623T12−21505406726124p6T14+45548573507p12T16+44311205p19T18+846032p24T20+1213p30T22+p36T24 1 + 1213 T^{2} + 846032 T^{4} + 44311205 p T^{6} + 45548573507 T^{8} - 21505406726124 T^{10} - 12859398267661623 T^{12} - 21505406726124 p^{6} T^{14} + 45548573507 p^{12} T^{16} + 44311205 p^{19} T^{18} + 846032 p^{24} T^{20} + 1213 p^{30} T^{22} + p^{36} T^{24}
11 1+2940T2+5572564T4+4606009814T6−2510834452120T8−15971978573781272T10−27825171402513079525T12−15971978573781272p6T14−2510834452120p12T16+4606009814p18T18+5572564p24T20+2940p30T22+p36T24 1 + 2940 T^{2} + 5572564 T^{4} + 4606009814 T^{6} - 2510834452120 T^{8} - 15971978573781272 T^{10} - 27825171402513079525 T^{12} - 15971978573781272 p^{6} T^{14} - 2510834452120 p^{12} T^{16} + 4606009814 p^{18} T^{18} + 5572564 p^{24} T^{20} + 2940 p^{30} T^{22} + p^{36} T^{24}
17 (1+90T−5468T2−393226T3+44785470T4+1125379354T5−211247539689T6+1125379354p3T7+44785470p6T8−393226p9T9−5468p12T10+90p15T11+p18T12)2 ( 1 + 90 T - 5468 T^{2} - 393226 T^{3} + 44785470 T^{4} + 1125379354 T^{5} - 211247539689 T^{6} + 1125379354 p^{3} T^{7} + 44785470 p^{6} T^{8} - 393226 p^{9} T^{9} - 5468 p^{12} T^{10} + 90 p^{15} T^{11} + p^{18} T^{12} )^{2}
19 1+15916T2+202761708T4+1133695405670T6+2795908662819632T8−48803075506997114688T10− 1 + 15916 T^{2} + 202761708 T^{4} + 1133695405670 T^{6} + 2795908662819632 T^{8} - 48803075506997114688 T^{10} - 44 ⁣⋯ ⁣5744\!\cdots\!57T12−48803075506997114688p6T14+2795908662819632p12T16+1133695405670p18T18+202761708p24T20+15916p30T22+p36T24 T^{12} - 48803075506997114688 p^{6} T^{14} + 2795908662819632 p^{12} T^{16} + 1133695405670 p^{18} T^{18} + 202761708 p^{24} T^{20} + 15916 p^{30} T^{22} + p^{36} T^{24}
23 (1−19T−17379T2+1795946T3+78601463T4−13530427071T5+857245646806T6−13530427071p3T7+78601463p6T8+1795946p9T9−17379p12T10−19p15T11+p18T12)2 ( 1 - 19 T - 17379 T^{2} + 1795946 T^{3} + 78601463 T^{4} - 13530427071 T^{5} + 857245646806 T^{6} - 13530427071 p^{3} T^{7} + 78601463 p^{6} T^{8} + 1795946 p^{9} T^{9} - 17379 p^{12} T^{10} - 19 p^{15} T^{11} + p^{18} T^{12} )^{2}
29 (1−101T+9670T2+2392827T3−136214727T4−58110677102T5+24435911693829T6−58110677102p3T7−136214727p6T8+2392827p9T9+9670p12T10−101p15T11+p18T12)2 ( 1 - 101 T + 9670 T^{2} + 2392827 T^{3} - 136214727 T^{4} - 58110677102 T^{5} + 24435911693829 T^{6} - 58110677102 p^{3} T^{7} - 136214727 p^{6} T^{8} + 2392827 p^{9} T^{9} + 9670 p^{12} T^{10} - 101 p^{15} T^{11} + p^{18} T^{12} )^{2}
31 (1−67693T2+2813159873T4−88764051088713T6+2813159873p6T8−67693p12T10+p18T12)2 ( 1 - 67693 T^{2} + 2813159873 T^{4} - 88764051088713 T^{6} + 2813159873 p^{6} T^{8} - 67693 p^{12} T^{10} + p^{18} T^{12} )^{2}
37 1+289120T2+48092940392T4+5518178251433198T6+ 1 + 289120 T^{2} + 48092940392 T^{4} + 5518178251433198 T^{6} + 48 ⁣⋯ ⁣7248\!\cdots\!72T8+ T^{8} + 33 ⁣⋯ ⁣8833\!\cdots\!88T10+ T^{10} + 18 ⁣⋯ ⁣4318\!\cdots\!43T12+ T^{12} + 33 ⁣⋯ ⁣8833\!\cdots\!88p6T14+ p^{6} T^{14} + 48 ⁣⋯ ⁣7248\!\cdots\!72p12T16+5518178251433198p18T18+48092940392p24T20+289120p30T22+p36T24 p^{12} T^{16} + 5518178251433198 p^{18} T^{18} + 48092940392 p^{24} T^{20} + 289120 p^{30} T^{22} + p^{36} T^{24}
41 1+72459T2−2942996285T4−784013353069900T6−21701918099899481623T8+ 1 + 72459 T^{2} - 2942996285 T^{4} - 784013353069900 T^{6} - 21701918099899481623 T^{8} + 16 ⁣⋯ ⁣7316\!\cdots\!73T10+ T^{10} + 25 ⁣⋯ ⁣9425\!\cdots\!94T12+ T^{12} + 16 ⁣⋯ ⁣7316\!\cdots\!73p6T14−21701918099899481623p12T16−784013353069900p18T18−2942996285p24T20+72459p30T22+p36T24 p^{6} T^{14} - 21701918099899481623 p^{12} T^{16} - 784013353069900 p^{18} T^{18} - 2942996285 p^{24} T^{20} + 72459 p^{30} T^{22} + p^{36} T^{24}
43 (1+1205T+731136T2+357194599T3+154200549383T4+53845297050762T5+15910224576330043T6+53845297050762p3T7+154200549383p6T8+357194599p9T9+731136p12T10+1205p15T11+p18T12)2 ( 1 + 1205 T + 731136 T^{2} + 357194599 T^{3} + 154200549383 T^{4} + 53845297050762 T^{5} + 15910224576330043 T^{6} + 53845297050762 p^{3} T^{7} + 154200549383 p^{6} T^{8} + 357194599 p^{9} T^{9} + 731136 p^{12} T^{10} + 1205 p^{15} T^{11} + p^{18} T^{12} )^{2}
47 (1−184924T2+13473005240T4−366518815821141T6+13473005240p6T8−184924p12T10+p18T12)2 ( 1 - 184924 T^{2} + 13473005240 T^{4} - 366518815821141 T^{6} + 13473005240 p^{6} T^{8} - 184924 p^{12} T^{10} + p^{18} T^{12} )^{2}
53 (1+1095T+839313T2+371911379T3+839313p3T4+1095p6T5+p9T6)4 ( 1 + 1095 T + 839313 T^{2} + 371911379 T^{3} + 839313 p^{3} T^{4} + 1095 p^{6} T^{5} + p^{9} T^{6} )^{4}
59 1+412807T2+70300999051T4+4687605950834692T6− 1 + 412807 T^{2} + 70300999051 T^{4} + 4687605950834692 T^{6} - 17 ⁣⋯ ⁣4317\!\cdots\!43T8− T^{8} - 91 ⁣⋯ ⁣7991\!\cdots\!79T10− T^{10} - 24 ⁣⋯ ⁣8224\!\cdots\!82T12− T^{12} - 91 ⁣⋯ ⁣7991\!\cdots\!79p6T14− p^{6} T^{14} - 17 ⁣⋯ ⁣4317\!\cdots\!43p12T16+4687605950834692p18T18+70300999051p24T20+412807p30T22+p36T24 p^{12} T^{16} + 4687605950834692 p^{18} T^{18} + 70300999051 p^{24} T^{20} + 412807 p^{30} T^{22} + p^{36} T^{24}
61 (1−1108T+195280T2−81146578T3+242730039476T4−86381104403572T5+569801340677051T6−86381104403572p3T7+242730039476p6T8−81146578p9T9+195280p12T10−1108p15T11+p18T12)2 ( 1 - 1108 T + 195280 T^{2} - 81146578 T^{3} + 242730039476 T^{4} - 86381104403572 T^{5} + 569801340677051 T^{6} - 86381104403572 p^{3} T^{7} + 242730039476 p^{6} T^{8} - 81146578 p^{9} T^{9} + 195280 p^{12} T^{10} - 1108 p^{15} T^{11} + p^{18} T^{12} )^{2}
67 1+754705T2+248140267820T4+44550675395447459T6− 1 + 754705 T^{2} + 248140267820 T^{4} + 44550675395447459 T^{6} - 48 ⁣⋯ ⁣8548\!\cdots\!85T8− T^{8} - 58 ⁣⋯ ⁣8058\!\cdots\!80T10− T^{10} - 27 ⁣⋯ ⁣4327\!\cdots\!43T12− T^{12} - 58 ⁣⋯ ⁣8058\!\cdots\!80p6T14− p^{6} T^{14} - 48 ⁣⋯ ⁣8548\!\cdots\!85p12T16+44550675395447459p18T18+248140267820p24T20+754705p30T22+p36T24 p^{12} T^{16} + 44550675395447459 p^{18} T^{18} + 248140267820 p^{24} T^{20} + 754705 p^{30} T^{22} + p^{36} T^{24}
71 1+1203945T2+861630964656T4+299768879179299207T6+ 1 + 1203945 T^{2} + 861630964656 T^{4} + 299768879179299207 T^{6} + 24 ⁣⋯ ⁣0724\!\cdots\!07T8− T^{8} - 41 ⁣⋯ ⁣8441\!\cdots\!84T10− T^{10} - 22 ⁣⋯ ⁣8322\!\cdots\!83T12− T^{12} - 41 ⁣⋯ ⁣8441\!\cdots\!84p6T14+ p^{6} T^{14} + 24 ⁣⋯ ⁣0724\!\cdots\!07p12T16+299768879179299207p18T18+861630964656p24T20+1203945p30T22+p36T24 p^{12} T^{16} + 299768879179299207 p^{18} T^{18} + 861630964656 p^{24} T^{20} + 1203945 p^{30} T^{22} + p^{36} T^{24}
73 (1−578397T2+340509020657T4−175733770047454529T6+340509020657p6T8−578397p12T10+p18T12)2 ( 1 - 578397 T^{2} + 340509020657 T^{4} - 175733770047454529 T^{6} + 340509020657 p^{6} T^{8} - 578397 p^{12} T^{10} + p^{18} T^{12} )^{2}
79 (1+1961T+2641773T2+2147763857T3+2641773p3T4+1961p6T5+p9T6)4 ( 1 + 1961 T + 2641773 T^{2} + 2147763857 T^{3} + 2641773 p^{3} T^{4} + 1961 p^{6} T^{5} + p^{9} T^{6} )^{4}
83 (1−3126181T2+4236994280213T4−3174148229464390881T6+4236994280213p6T8−3126181p12T10+p18T12)2 ( 1 - 3126181 T^{2} + 4236994280213 T^{4} - 3174148229464390881 T^{6} + 4236994280213 p^{6} T^{8} - 3126181 p^{12} T^{10} + p^{18} T^{12} )^{2}
89 1+3632513T2+7360263332404T4+10561614224132377739T6+ 1 + 3632513 T^{2} + 7360263332404 T^{4} + 10561614224132377739 T^{6} + 11 ⁣⋯ ⁣7111\!\cdots\!71T8+ T^{8} + 10 ⁣⋯ ⁣2810\!\cdots\!28T10+ T^{10} + 83 ⁣⋯ ⁣9783\!\cdots\!97T12+ T^{12} + 10 ⁣⋯ ⁣2810\!\cdots\!28p6T14+ p^{6} T^{14} + 11 ⁣⋯ ⁣7111\!\cdots\!71p12T16+10561614224132377739p18T18+7360263332404p24T20+3632513p30T22+p36T24 p^{12} T^{16} + 10561614224132377739 p^{18} T^{18} + 7360263332404 p^{24} T^{20} + 3632513 p^{30} T^{22} + p^{36} T^{24}
97 1+3266797T2+5378593419936T4+5916317576309886611T6+ 1 + 3266797 T^{2} + 5378593419936 T^{4} + 5916317576309886611 T^{6} + 50 ⁣⋯ ⁣6750\!\cdots\!67T8+ T^{8} + 35 ⁣⋯ ⁣8435\!\cdots\!84T10+ T^{10} + 27 ⁣⋯ ⁣9327\!\cdots\!93T12+ T^{12} + 35 ⁣⋯ ⁣8435\!\cdots\!84p6T14+ p^{6} T^{14} + 50 ⁣⋯ ⁣6750\!\cdots\!67p12T16+5916317576309886611p18T18+5378593419936p24T20+3266797p30T22+p36T24 p^{12} T^{16} + 5916317576309886611 p^{18} T^{18} + 5378593419936 p^{24} T^{20} + 3266797 p^{30} T^{22} + p^{36} T^{24}
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   L(s)=∏p ∏j=124(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.26497898307573854396602347696, −3.25961829442592949411396701286, −3.03000248180092872780894719604, −2.98532768305944774114661758752, −2.98105278733210415336166583976, −2.96739015943463796250869315628, −2.71646049329439524874690439539, −2.59856449349083383252209087232, −2.57754250324296734544808974412, −2.51670539696914686524641413787, −2.30101256485813820157234829878, −2.12550085636595206337915687446, −1.85387196551216425189324708488, −1.69789291561701999109720965013, −1.68115457801384361916567776499, −1.57672033931221252196939285676, −1.52852970341677321894936732830, −1.33230835016512939445078442583, −1.32575760091362613998215881256, −1.30188328656520887090271364335, −1.23486323533708843593078024304, −0.72501828456551874048616655719, −0.52383535353422078080396923563, −0.15867713882680696411996591993, −0.07303609752524847649026298262, 0.07303609752524847649026298262, 0.15867713882680696411996591993, 0.52383535353422078080396923563, 0.72501828456551874048616655719, 1.23486323533708843593078024304, 1.30188328656520887090271364335, 1.32575760091362613998215881256, 1.33230835016512939445078442583, 1.52852970341677321894936732830, 1.57672033931221252196939285676, 1.68115457801384361916567776499, 1.69789291561701999109720965013, 1.85387196551216425189324708488, 2.12550085636595206337915687446, 2.30101256485813820157234829878, 2.51670539696914686524641413787, 2.57754250324296734544808974412, 2.59856449349083383252209087232, 2.71646049329439524874690439539, 2.96739015943463796250869315628, 2.98105278733210415336166583976, 2.98532768305944774114661758752, 3.03000248180092872780894719604, 3.25961829442592949411396701286, 3.26497898307573854396602347696

Graph of the ZZ-function along the critical line

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