Properties

Label 12-74e6-1.1-c1e6-0-1
Degree $12$
Conductor $164206490176$
Sign $1$
Analytic cond. $0.0425650$
Root an. cond. $0.768695$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 3·5-s + 6·7-s + 8-s + 3·9-s − 9·11-s − 9·15-s + 9·19-s − 18·21-s − 15·23-s − 3·24-s + 15·25-s + 27-s − 18·31-s + 27·33-s + 18·35-s + 9·37-s + 3·40-s + 6·41-s + 12·43-s + 9·45-s − 3·47-s + 18·49-s − 18·53-s − 27·55-s + 6·56-s − 27·57-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.34·5-s + 2.26·7-s + 0.353·8-s + 9-s − 2.71·11-s − 2.32·15-s + 2.06·19-s − 3.92·21-s − 3.12·23-s − 0.612·24-s + 3·25-s + 0.192·27-s − 3.23·31-s + 4.70·33-s + 3.04·35-s + 1.47·37-s + 0.474·40-s + 0.937·41-s + 1.82·43-s + 1.34·45-s − 0.437·47-s + 18/7·49-s − 2.47·53-s − 3.64·55-s + 0.801·56-s − 3.57·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 37^{6}\)
Sign: $1$
Analytic conductor: \(0.0425650\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{74} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 37^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4980184850\)
\(L(\frac12)\) \(\approx\) \(0.4980184850\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{3} + T^{6} \)
37 \( 1 - 9 T + 54 T^{2} - 305 T^{3} + 54 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
good3 \( 1 + p T + 2 p T^{2} + 8 T^{3} + p T^{4} - 7 p T^{5} - 53 T^{6} - 7 p^{2} T^{7} + p^{3} T^{8} + 8 p^{3} T^{9} + 2 p^{5} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 - 3 T - 6 T^{2} + 38 T^{3} - 51 T^{4} - 117 T^{5} + 581 T^{6} - 117 p T^{7} - 51 p^{2} T^{8} + 38 p^{3} T^{9} - 6 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 6 T + 18 T^{2} - 51 T^{3} + 99 T^{4} - 69 T^{5} - 19 T^{6} - 69 p T^{7} + 99 p^{2} T^{8} - 51 p^{3} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 9 T + 24 T^{2} + 83 T^{3} + 687 T^{4} + 2058 T^{5} + 3347 T^{6} + 2058 p T^{7} + 687 p^{2} T^{8} + 83 p^{3} T^{9} + 24 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 36 T^{2} + 27 T^{3} + 63 p T^{4} + 549 T^{5} + 12977 T^{6} + 549 p T^{7} + 63 p^{3} T^{8} + 27 p^{3} T^{9} + 36 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 126 T^{3} + 10963 T^{6} + 126 p^{3} T^{9} + p^{6} T^{12} \)
19 \( 1 - 9 T + 63 T^{2} - 295 T^{3} + 1188 T^{4} - 3510 T^{5} + 11397 T^{6} - 3510 p T^{7} + 1188 p^{2} T^{8} - 295 p^{3} T^{9} + 63 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 15 T + 102 T^{2} + 459 T^{3} + 1905 T^{4} + 8304 T^{5} + 37591 T^{6} + 8304 p T^{7} + 1905 p^{2} T^{8} + 459 p^{3} T^{9} + 102 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 24 T^{2} - 342 T^{3} - 120 T^{4} + 4104 T^{5} + 61315 T^{6} + 4104 p T^{7} - 120 p^{2} T^{8} - 342 p^{3} T^{9} - 24 p^{4} T^{10} + p^{6} T^{12} \)
31 \( ( 1 + 9 T + 117 T^{2} + 575 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 - 6 T + 36 T^{2} + 54 T^{3} - 288 T^{4} + 5232 T^{5} + 44551 T^{6} + 5232 p T^{7} - 288 p^{2} T^{8} + 54 p^{3} T^{9} + 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
43 \( ( 1 - 6 T + 48 T^{2} - 49 T^{3} + 48 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 3 T - 87 T^{2} - 310 T^{3} + 81 p T^{4} + 8259 T^{5} - 155986 T^{6} + 8259 p T^{7} + 81 p^{3} T^{8} - 310 p^{3} T^{9} - 87 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 18 T + 144 T^{2} + 549 T^{3} - 4113 T^{4} - 79569 T^{5} - 665387 T^{6} - 79569 p T^{7} - 4113 p^{2} T^{8} + 549 p^{3} T^{9} + 144 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 6 T + 120 T^{2} + 83 T^{3} + 3477 T^{4} - 46503 T^{5} + 35645 T^{6} - 46503 p T^{7} + 3477 p^{2} T^{8} + 83 p^{3} T^{9} + 120 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 12 T + 48 T^{2} + 220 T^{3} - 4320 T^{4} - 57240 T^{5} - 269733 T^{6} - 57240 p T^{7} - 4320 p^{2} T^{8} + 220 p^{3} T^{9} + 48 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 3 T - 36 T^{2} - 1140 T^{3} - 4194 T^{4} + 29397 T^{5} + 912401 T^{6} + 29397 p T^{7} - 4194 p^{2} T^{8} - 1140 p^{3} T^{9} - 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 6 T - 144 T^{2} - 1080 T^{3} + 2700 T^{4} + 46464 T^{5} + 353557 T^{6} + 46464 p T^{7} + 2700 p^{2} T^{8} - 1080 p^{3} T^{9} - 144 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( ( 1 + 18 T + 306 T^{2} + 2681 T^{3} + 306 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( 1 - 30 T + 360 T^{2} - 2028 T^{3} + 5490 T^{4} - 36318 T^{5} + 477341 T^{6} - 36318 p T^{7} + 5490 p^{2} T^{8} - 2028 p^{3} T^{9} + 360 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 6 T - 12 T^{2} + 719 T^{3} - 5007 T^{4} - 73899 T^{5} + 904205 T^{6} - 73899 p T^{7} - 5007 p^{2} T^{8} + 719 p^{3} T^{9} - 12 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 33 T + 516 T^{2} + 4394 T^{3} + 6204 T^{4} - 396549 T^{5} - 5838061 T^{6} - 396549 p T^{7} + 6204 p^{2} T^{8} + 4394 p^{3} T^{9} + 516 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 42 T + 897 T^{2} - 14982 T^{3} + 217518 T^{4} - 2621634 T^{5} + 27146693 T^{6} - 2621634 p T^{7} + 217518 p^{2} T^{8} - 14982 p^{3} T^{9} + 897 p^{4} T^{10} - 42 p^{5} T^{11} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.174048153995259133981210448840, −7.898322386672053572906011818409, −7.81507301990169446241714833891, −7.81205207965268755664204507915, −7.55697696988713929644103738205, −7.27587729921328182570226523607, −7.25307361417278530826443201261, −6.84975231250723620386439949545, −6.10008485594824768482219715413, −6.06633479311166729227181358383, −6.04520785006568581864829579046, −5.88364167419000677108053926181, −5.66185389277856697238812700976, −5.25838212421876824016455278450, −4.96917904069463441676932877451, −4.88238000245229935233914848553, −4.85644310162187915331938115454, −4.61646962993621454086158432368, −3.93218650718567271627324299217, −3.76372049722118746170289440162, −3.06339778718228971833146906353, −2.63843301892992575746397326465, −2.49377108056394432966295850798, −1.67447562853816073528053822664, −1.59013270866072073062704600361, 1.59013270866072073062704600361, 1.67447562853816073528053822664, 2.49377108056394432966295850798, 2.63843301892992575746397326465, 3.06339778718228971833146906353, 3.76372049722118746170289440162, 3.93218650718567271627324299217, 4.61646962993621454086158432368, 4.85644310162187915331938115454, 4.88238000245229935233914848553, 4.96917904069463441676932877451, 5.25838212421876824016455278450, 5.66185389277856697238812700976, 5.88364167419000677108053926181, 6.04520785006568581864829579046, 6.06633479311166729227181358383, 6.10008485594824768482219715413, 6.84975231250723620386439949545, 7.25307361417278530826443201261, 7.27587729921328182570226523607, 7.55697696988713929644103738205, 7.81205207965268755664204507915, 7.81507301990169446241714833891, 7.898322386672053572906011818409, 8.174048153995259133981210448840

Graph of the $Z$-function along the critical line

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