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Properties

Label 12-338e6-1.1-c3e6-0-3
Degree 1212
Conductor 1.491×10151.491\times 10^{15}
Sign 11
Analytic cond. 6.29066×1076.29066\times 10^{7}
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  + 6·2-s + 12·3-s + 12·4-s + 24·5-s + 72·6-s + 27·7-s − 16·8-s + 108·9-s + 144·10-s − 82·11-s + 144·12-s + 162·14-s + 288·15-s − 144·16-s + 90·17-s + 648·18-s − 130·19-s + 288·20-s + 324·21-s − 492·22-s − 19·23-s − 192·24-s − 148·25-s + 890·27-s + 324·28-s + 101·29-s + 1.72e3·30-s + ⋯
L(s)  = 1  + 2.12·2-s + 2.30·3-s + 3/2·4-s + 2.14·5-s + 4.89·6-s + 1.45·7-s − 0.707·8-s + 4·9-s + 4.55·10-s − 2.24·11-s + 3.46·12-s + 3.09·14-s + 4.95·15-s − 9/4·16-s + 1.28·17-s + 8.48·18-s − 1.56·19-s + 3.21·20-s + 3.36·21-s − 4.76·22-s − 0.172·23-s − 1.63·24-s − 1.18·25-s + 6.34·27-s + 2.18·28-s + 0.646·29-s + 10.5·30-s + ⋯

Functional equation

Λ(s)=((26⋅1312)s/2 ΓC(s)6 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((26⋅1312)s/2 ΓC(s+3/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 26⋅13122^{6} \cdot 13^{12}
Sign: 11
Analytic conductor: 6.29066×1076.29066\times 10^{7}
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: (12, 26⋅1312, ( :[3/2]6), 1)(12,\ 2^{6} \cdot 13^{12} ,\ ( \ : [3/2]^{6} ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 75.8379290075.83792900
L(12)L(\frac12) ≈\approx 75.8379290075.83792900
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−pT+p2T2)3 ( 1 - p T + p^{2} T^{2} )^{3}
13 1 1
good3 1−4pT+4p2T2−26T3+376pT4−272p3T5+22687T6−272p6T7+376p7T8−26p9T9+4p14T10−4p16T11+p18T12 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}
5 (1−12T+58pT2−2231T3+58p4T4−12p6T5+p9T6)2 ( 1 - 12 T + 58 p T^{2} - 2231 T^{3} + 58 p^{4} T^{4} - 12 p^{6} T^{5} + p^{9} T^{6} )^{2}
7 1−27T−242T2+1459pT3+117983T4−2878426T5+3332855T6−2878426p3T7+117983p6T8+1459p10T9−242p12T10−27p15T11+p18T12 1 - 27 T - 242 T^{2} + 1459 p T^{3} + 117983 T^{4} - 2878426 T^{5} + 3332855 T^{6} - 2878426 p^{3} T^{7} + 117983 p^{6} T^{8} + 1459 p^{10} T^{9} - 242 p^{12} T^{10} - 27 p^{15} T^{11} + p^{18} T^{12}
11 1+82T+172pT2−7922T3+346846T4+66936450T5+2560930403T6+66936450p3T7+346846p6T8−7922p9T9+172p13T10+82p15T11+p18T12 1 + 82 T + 172 p T^{2} - 7922 T^{3} + 346846 T^{4} + 66936450 T^{5} + 2560930403 T^{6} + 66936450 p^{3} T^{7} + 346846 p^{6} T^{8} - 7922 p^{9} T^{9} + 172 p^{13} T^{10} + 82 p^{15} T^{11} + p^{18} T^{12}
17 1−90T−5468T2+393226T3+44785470T4−1125379354T5−211247539689T6−1125379354p3T7+44785470p6T8+393226p9T9−5468p12T10−90p15T11+p18T12 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}
19 1+130T+492T2−1280422T3−65195038T4+4858964058T5+916633832443T6+4858964058p3T7−65195038p6T8−1280422p9T9+492p12T10+130p15T11+p18T12 1 + 130 T + 492 T^{2} - 1280422 T^{3} - 65195038 T^{4} + 4858964058 T^{5} + 916633832443 T^{6} + 4858964058 p^{3} T^{7} - 65195038 p^{6} T^{8} - 1280422 p^{9} T^{9} + 492 p^{12} T^{10} + 130 p^{15} T^{11} + p^{18} T^{12}
23 1+19T−17379T2−1795946T3+78601463T4+13530427071T5+857245646806T6+13530427071p3T7+78601463p6T8−1795946p9T9−17379p12T10+19p15T11+p18T12 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}
29 1−101T+9670T2+2392827T3−136214727T4−58110677102T5+24435911693829T6−58110677102p3T7−136214727p6T8+2392827p9T9+9670p12T10−101p15T11+p18T12 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}
31 (1−519T+168527T2−34325015T3+168527p3T4−519p6T5+p9T6)2 ( 1 - 519 T + 168527 T^{2} - 34325015 T^{3} + 168527 p^{3} T^{4} - 519 p^{6} T^{5} + p^{9} T^{6} )^{2}
37 1+84T−141032T2−4641854T3+13711541948T4+220099751228T5−796061158324453T6+220099751228p3T7+13711541948p6T8−4641854p9T9−141032p12T10+84p15T11+p18T12 1 + 84 T - 141032 T^{2} - 4641854 T^{3} + 13711541948 T^{4} + 220099751228 T^{5} - 796061158324453 T^{6} + 220099751228 p^{3} T^{7} + 13711541948 p^{6} T^{8} - 4641854 p^{9} T^{9} - 141032 p^{12} T^{10} + 84 p^{15} T^{11} + p^{18} T^{12}
41 1+187T−18745T2−1163696T3−1864796807T4−195267953691T5+321213047237726T6−195267953691p3T7−1864796807p6T8−1163696p9T9−18745p12T10+187p15T11+p18T12 1 + 187 T - 18745 T^{2} - 1163696 T^{3} - 1864796807 T^{4} - 195267953691 T^{5} + 321213047237726 T^{6} - 195267953691 p^{3} T^{7} - 1864796807 p^{6} T^{8} - 1163696 p^{9} T^{9} - 18745 p^{12} T^{10} + 187 p^{15} T^{11} + p^{18} T^{12}
43 1−1205T+731136T2−357194599T3+154200549383T4−53845297050762T5+15910224576330043T6−53845297050762p3T7+154200549383p6T8−357194599p9T9+731136p12T10−1205p15T11+p18T12 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}
47 (1+536T+236110T2+85169985T3+236110p3T4+536p6T5+p9T6)2 ( 1 + 536 T + 236110 T^{2} + 85169985 T^{3} + 236110 p^{3} T^{4} + 536 p^{6} T^{5} + p^{9} T^{6} )^{2}
53 (1+1095T+839313T2+371911379T3+839313p3T4+1095p6T5+p9T6)2 ( 1 + 1095 T + 839313 T^{2} + 371911379 T^{3} + 839313 p^{3} T^{4} + 1095 p^{6} T^{5} + p^{9} T^{6} )^{2}
59 1−1413T+791881T2−395784094T3+4760265513pT4−147325234570781T5+61745012937499398T6−147325234570781p3T7+4760265513p7T8−395784094p9T9+791881p12T10−1413p15T11+p18T12 1 - 1413 T + 791881 T^{2} - 395784094 T^{3} + 4760265513 p T^{4} - 147325234570781 T^{5} + 61745012937499398 T^{6} - 147325234570781 p^{3} T^{7} + 4760265513 p^{7} T^{8} - 395784094 p^{9} T^{9} + 791881 p^{12} T^{10} - 1413 p^{15} T^{11} + p^{18} T^{12}
61 1−1108T+195280T2−81146578T3+242730039476T4−86381104403572T5+569801340677051T6−86381104403572p3T7+242730039476p6T8−81146578p9T9+195280p12T10−1108p15T11+p18T12 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}
67 1−1605T+910660T2−521034463T3+545679629225T4−329190135263360T5+144884674066779755T6−329190135263360p3T7+545679629225p6T8−521034463p9T9+910660p12T10−1605p15T11+p18T12 1 - 1605 T + 910660 T^{2} - 521034463 T^{3} + 545679629225 T^{4} - 329190135263360 T^{5} + 144884674066779755 T^{6} - 329190135263360 p^{3} T^{7} + 545679629225 p^{6} T^{8} - 521034463 p^{9} T^{9} + 910660 p^{12} T^{10} - 1605 p^{15} T^{11} + p^{18} T^{12}
71 1−909T−188832T2+363557137T3+82513282683T4−108794514090066T5+30677817845635031T6−108794514090066p3T7+82513282683p6T8+363557137p9T9−188832p12T10−909p15T11+p18T12 1 - 909 T - 188832 T^{2} + 363557137 T^{3} + 82513282683 T^{4} - 108794514090066 T^{5} + 30677817845635031 T^{6} - 108794514090066 p^{3} T^{7} + 82513282683 p^{6} T^{8} + 363557137 p^{9} T^{9} - 188832 p^{12} T^{10} - 909 p^{15} T^{11} + p^{18} T^{12}
73 (1−287T+330383T2−44761521T3+330383p3T4−287p6T5+p9T6)2 ( 1 - 287 T + 330383 T^{2} - 44761521 T^{3} + 330383 p^{3} T^{4} - 287 p^{6} T^{5} + p^{9} T^{6} )^{2}
79 (1+1961T+2641773T2+2147763857T3+2641773p3T4+1961p6T5+p9T6)2 ( 1 + 1961 T + 2641773 T^{2} + 2147763857 T^{3} + 2641773 p^{3} T^{4} + 1961 p^{6} T^{5} + p^{9} T^{6} )^{2}
83 (1−191T+1581331T2−188435781T3+1581331p3T4−191p6T5+p9T6)2 ( 1 - 191 T + 1581331 T^{2} - 188435781 T^{3} + 1581331 p^{3} T^{4} - 191 p^{6} T^{5} + p^{9} T^{6} )^{2}
89 1+1091T−1221116T2−467120403T3+2424941163801T4+568092710973530T5−1590782774768604519T6+568092710973530p3T7+2424941163801p6T8−467120403p9T9−1221116p12T10+1091p15T11+p18T12 1 + 1091 T - 1221116 T^{2} - 467120403 T^{3} + 2424941163801 T^{4} + 568092710973530 T^{5} - 1590782774768604519 T^{6} + 568092710973530 p^{3} T^{7} + 2424941163801 p^{6} T^{8} - 467120403 p^{9} T^{9} - 1221116 p^{12} T^{10} + 1091 p^{15} T^{11} + p^{18} T^{12}
97 1−947T−1184994T2+1028052401T3+1013625696203T4−312418983925740T5−932711772529957343T6−312418983925740p3T7+1013625696203p6T8+1028052401p9T9−1184994p12T10−947p15T11+p18T12 1 - 947 T - 1184994 T^{2} + 1028052401 T^{3} + 1013625696203 T^{4} - 312418983925740 T^{5} - 932711772529957343 T^{6} - 312418983925740 p^{3} T^{7} + 1013625696203 p^{6} T^{8} + 1028052401 p^{9} T^{9} - 1184994 p^{12} T^{10} - 947 p^{15} T^{11} + p^{18} T^{12}
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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

−5.99497966750363648610796196676, −5.33616711429272316411229581420, −5.29163182565577810640190693331, −5.09629228600917135600808059315, −4.98082721206099078522670748968, −4.92444598129373887976544924042, −4.85033077413676610820447171656, −4.37330058479914445743904206286, −4.12613321646576111564151737233, −4.05012858621967743065775819574, −4.03806535787713110397928422795, −3.78404999486496844735709459735, −3.59819301963575328488713784598, −2.99299971755671735902898490000, −2.77958193027309022730476705826, −2.70555808031958039760688698305, −2.60061189598415159541384975493, −2.52171916662755754108608889458, −2.37777608752468918341659120484, −1.92554445152553832971210116648, −1.59115118833607130762161677267, −1.45869956145610847792304300067, −1.18406414129596056281812996160, −0.814021371173117294748187586922, −0.29577942596693607327655230950, 0.29577942596693607327655230950, 0.814021371173117294748187586922, 1.18406414129596056281812996160, 1.45869956145610847792304300067, 1.59115118833607130762161677267, 1.92554445152553832971210116648, 2.37777608752468918341659120484, 2.52171916662755754108608889458, 2.60061189598415159541384975493, 2.70555808031958039760688698305, 2.77958193027309022730476705826, 2.99299971755671735902898490000, 3.59819301963575328488713784598, 3.78404999486496844735709459735, 4.03806535787713110397928422795, 4.05012858621967743065775819574, 4.12613321646576111564151737233, 4.37330058479914445743904206286, 4.85033077413676610820447171656, 4.92444598129373887976544924042, 4.98082721206099078522670748968, 5.09629228600917135600808059315, 5.29163182565577810640190693331, 5.33616711429272316411229581420, 5.99497966750363648610796196676

Graph of the ZZ-function along the critical line

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