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Properties

Label 12-7448e6-1.1-c1e6-0-0
Degree $12$
Conductor $1.707\times 10^{23}$
Sign $1$
Analytic cond. $4.42487\times 10^{10}$
Root an. cond. $7.71184$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 5-s − 3·9-s + 3·11-s + 6·13-s + 3·15-s − 2·17-s − 6·19-s + 2·23-s − 12·25-s − 19·27-s − 29-s + 14·31-s + 9·33-s − 7·37-s + 18·39-s − 21·41-s + 43-s − 3·45-s + 23·47-s − 6·51-s − 15·53-s + 3·55-s − 18·57-s + 25·59-s + 21·61-s + 6·65-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.447·5-s − 9-s + 0.904·11-s + 1.66·13-s + 0.774·15-s − 0.485·17-s − 1.37·19-s + 0.417·23-s − 2.39·25-s − 3.65·27-s − 0.185·29-s + 2.51·31-s + 1.56·33-s − 1.15·37-s + 2.88·39-s − 3.27·41-s + 0.152·43-s − 0.447·45-s + 3.35·47-s − 0.840·51-s − 2.06·53-s + 0.404·55-s − 2.38·57-s + 3.25·59-s + 2.68·61-s + 0.744·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 7^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(4.42487\times 10^{10}\)
Root analytic conductor: \(7.71184\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 7^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(13.56039887\)
\(L(\frac12)\) \(\approx\) \(13.56039887\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( ( 1 + T )^{6} \)
good3 \( 1 - p T + 4 p T^{2} - 26 T^{3} + 65 T^{4} - 118 T^{5} + 235 T^{6} - 118 p T^{7} + 65 p^{2} T^{8} - 26 p^{3} T^{9} + 4 p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
5 \( 1 - T + 13 T^{2} - T^{3} + 94 T^{4} - 13 T^{5} + 593 T^{6} - 13 p T^{7} + 94 p^{2} T^{8} - p^{3} T^{9} + 13 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 3 T + 38 T^{2} - 4 p T^{3} + 587 T^{4} - 16 p T^{5} + 6955 T^{6} - 16 p^{2} T^{7} + 587 p^{2} T^{8} - 4 p^{4} T^{9} + 38 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 6 T + 38 T^{2} - 196 T^{3} + 1094 T^{4} - 4118 T^{5} + 16074 T^{6} - 4118 p T^{7} + 1094 p^{2} T^{8} - 196 p^{3} T^{9} + 38 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 2 T + 47 T^{2} + 32 T^{3} + 686 T^{4} - 936 T^{5} + 6132 T^{6} - 936 p T^{7} + 686 p^{2} T^{8} + 32 p^{3} T^{9} + 47 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 2 T + 60 T^{2} - 90 T^{3} + 2128 T^{4} - 2098 T^{5} + 51882 T^{6} - 2098 p T^{7} + 2128 p^{2} T^{8} - 90 p^{3} T^{9} + 60 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + T + 84 T^{2} + 292 T^{3} + 4287 T^{4} + 430 p T^{5} + 165657 T^{6} + 430 p^{2} T^{7} + 4287 p^{2} T^{8} + 292 p^{3} T^{9} + 84 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 14 T + 183 T^{2} - 1574 T^{3} + 13270 T^{4} - 85522 T^{5} + 533744 T^{6} - 85522 p T^{7} + 13270 p^{2} T^{8} - 1574 p^{3} T^{9} + 183 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 7 T + 195 T^{2} + 1229 T^{3} + 16632 T^{4} + 88507 T^{5} + 798157 T^{6} + 88507 p T^{7} + 16632 p^{2} T^{8} + 1229 p^{3} T^{9} + 195 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 21 T + 240 T^{2} + 2384 T^{3} + 22277 T^{4} + 169288 T^{5} + 1114543 T^{6} + 169288 p T^{7} + 22277 p^{2} T^{8} + 2384 p^{3} T^{9} + 240 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - T + 207 T^{2} - 163 T^{3} + 19507 T^{4} - 12006 T^{5} + 1069658 T^{6} - 12006 p T^{7} + 19507 p^{2} T^{8} - 163 p^{3} T^{9} + 207 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 23 T + 441 T^{2} - 5425 T^{3} + 59394 T^{4} - 499907 T^{5} + 3837489 T^{6} - 499907 p T^{7} + 59394 p^{2} T^{8} - 5425 p^{3} T^{9} + 441 p^{4} T^{10} - 23 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 15 T + 282 T^{2} + 2954 T^{3} + 33121 T^{4} + 262260 T^{5} + 2228853 T^{6} + 262260 p T^{7} + 33121 p^{2} T^{8} + 2954 p^{3} T^{9} + 282 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 25 T + 575 T^{2} - 8231 T^{3} + 105840 T^{4} - 1022633 T^{5} + 8896649 T^{6} - 1022633 p T^{7} + 105840 p^{2} T^{8} - 8231 p^{3} T^{9} + 575 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 21 T + 377 T^{2} - 5369 T^{3} + 60746 T^{4} - 593905 T^{5} + 5053473 T^{6} - 593905 p T^{7} + 60746 p^{2} T^{8} - 5369 p^{3} T^{9} + 377 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 2 T + 149 T^{2} - 174 T^{3} + 11080 T^{4} - 10116 T^{5} + 752920 T^{6} - 10116 p T^{7} + 11080 p^{2} T^{8} - 174 p^{3} T^{9} + 149 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 13 T + 427 T^{2} - 4217 T^{3} + 75440 T^{4} - 572805 T^{5} + 7125101 T^{6} - 572805 p T^{7} + 75440 p^{2} T^{8} - 4217 p^{3} T^{9} + 427 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 2 T + 193 T^{2} + 1056 T^{3} + 24130 T^{4} + 124600 T^{5} + 2172088 T^{6} + 124600 p T^{7} + 24130 p^{2} T^{8} + 1056 p^{3} T^{9} + 193 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 17 T + 229 T^{2} + 2807 T^{3} + 30463 T^{4} + 298342 T^{5} + 2905158 T^{6} + 298342 p T^{7} + 30463 p^{2} T^{8} + 2807 p^{3} T^{9} + 229 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 4 T + 365 T^{2} - 1480 T^{3} + 64106 T^{4} - 229140 T^{5} + 6723468 T^{6} - 229140 p T^{7} + 64106 p^{2} T^{8} - 1480 p^{3} T^{9} + 365 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + T + 155 T^{2} - 395 T^{3} + 19563 T^{4} - 54414 T^{5} + 2076002 T^{6} - 54414 p T^{7} + 19563 p^{2} T^{8} - 395 p^{3} T^{9} + 155 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 13 T + 513 T^{2} + 4993 T^{3} + 113816 T^{4} + 874477 T^{5} + 14326867 T^{6} + 874477 p T^{7} + 113816 p^{2} T^{8} + 4993 p^{3} T^{9} + 513 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.04353688401891213443012804667, −3.75691997088265340831970380138, −3.67759925714835914760876808775, −3.62627828824985047544388995946, −3.47969302768355217348861589041, −3.36999502093772818612884992383, −3.34070685697253255449727487013, −2.81679607355224842747371682823, −2.81327218714800778804683842661, −2.80972643845985062692944197964, −2.62546678405970884981339632285, −2.60777959893974206024827491086, −2.57894434228785023446464824389, −2.02297441426854016714803530106, −1.95405888892133897879923615743, −1.94268461900725684328179818425, −1.91514517869104051945203381065, −1.66597776783543413463773721574, −1.55389604807115672466591522309, −1.29523553213946620977412569662, −0.937438010895845612405508485826, −0.71909367552602301961286830506, −0.68998048756600089271000188327, −0.44884530169119687691162962628, −0.22944574605280368112405544904, 0.22944574605280368112405544904, 0.44884530169119687691162962628, 0.68998048756600089271000188327, 0.71909367552602301961286830506, 0.937438010895845612405508485826, 1.29523553213946620977412569662, 1.55389604807115672466591522309, 1.66597776783543413463773721574, 1.91514517869104051945203381065, 1.94268461900725684328179818425, 1.95405888892133897879923615743, 2.02297441426854016714803530106, 2.57894434228785023446464824389, 2.60777959893974206024827491086, 2.62546678405970884981339632285, 2.80972643845985062692944197964, 2.81327218714800778804683842661, 2.81679607355224842747371682823, 3.34070685697253255449727487013, 3.36999502093772818612884992383, 3.47969302768355217348861589041, 3.62627828824985047544388995946, 3.67759925714835914760876808775, 3.75691997088265340831970380138, 4.04353688401891213443012804667

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

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