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Properties

Label 12-5160e6-1.1-c1e6-0-2
Degree $12$
Conductor $1.888\times 10^{22}$
Sign $1$
Analytic cond. $4.89283\times 10^{9}$
Root an. cond. $6.41894$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 6·5-s − 6·7-s + 21·9-s + 5·11-s − 3·13-s − 36·15-s − 3·17-s − 6·19-s − 36·21-s − 7·23-s + 21·25-s + 56·27-s − 4·29-s − 17·31-s + 30·33-s + 36·35-s − 14·37-s − 18·39-s − 5·41-s − 6·43-s − 126·45-s − 20·47-s − 18·51-s − 13·53-s − 30·55-s − 36·57-s + ⋯
L(s)  = 1  + 3.46·3-s − 2.68·5-s − 2.26·7-s + 7·9-s + 1.50·11-s − 0.832·13-s − 9.29·15-s − 0.727·17-s − 1.37·19-s − 7.85·21-s − 1.45·23-s + 21/5·25-s + 10.7·27-s − 0.742·29-s − 3.05·31-s + 5.22·33-s + 6.08·35-s − 2.30·37-s − 2.88·39-s − 0.780·41-s − 0.914·43-s − 18.7·45-s − 2.91·47-s − 2.52·51-s − 1.78·53-s − 4.04·55-s − 4.76·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6} \cdot 43^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{6} \cdot 5^{6} \cdot 43^{6}\)
Sign: $1$
Analytic conductor: \(4.89283\times 10^{9}\)
Root analytic conductor: \(6.41894\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{18} \cdot 3^{6} \cdot 5^{6} \cdot 43^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 - T )^{6} \)
5 \( ( 1 + T )^{6} \)
43 \( ( 1 + T )^{6} \)
good7 \( 1 + 6 T + 36 T^{2} + 136 T^{3} + 519 T^{4} + 1514 T^{5} + 4488 T^{6} + 1514 p T^{7} + 519 p^{2} T^{8} + 136 p^{3} T^{9} + 36 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 5 T + 47 T^{2} - 216 T^{3} + 1101 T^{4} - 35 p^{2} T^{5} + 15326 T^{6} - 35 p^{3} T^{7} + 1101 p^{2} T^{8} - 216 p^{3} T^{9} + 47 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 3 T + 27 T^{2} + 46 T^{3} + 369 T^{4} + 11 p T^{5} + 3126 T^{6} + 11 p^{2} T^{7} + 369 p^{2} T^{8} + 46 p^{3} T^{9} + 27 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 3 T + 45 T^{2} + 28 T^{3} + 991 T^{4} + 189 T^{5} + 20694 T^{6} + 189 p T^{7} + 991 p^{2} T^{8} + 28 p^{3} T^{9} + 45 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 6 T + 42 T^{2} + 296 T^{3} + 1851 T^{4} + 8594 T^{5} + 38740 T^{6} + 8594 p T^{7} + 1851 p^{2} T^{8} + 296 p^{3} T^{9} + 42 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 7 T + 49 T^{2} + 226 T^{3} + 1491 T^{4} + 7527 T^{5} + 45142 T^{6} + 7527 p T^{7} + 1491 p^{2} T^{8} + 226 p^{3} T^{9} + 49 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 4 T + 74 T^{2} + 148 T^{3} + 2951 T^{4} + 1560 T^{5} + 80556 T^{6} + 1560 p T^{7} + 2951 p^{2} T^{8} + 148 p^{3} T^{9} + 74 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 17 T + 221 T^{2} + 2206 T^{3} + 18267 T^{4} + 127377 T^{5} + 763790 T^{6} + 127377 p T^{7} + 18267 p^{2} T^{8} + 2206 p^{3} T^{9} + 221 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 14 T + 258 T^{2} + 2486 T^{3} + 25639 T^{4} + 180564 T^{5} + 1292060 T^{6} + 180564 p T^{7} + 25639 p^{2} T^{8} + 2486 p^{3} T^{9} + 258 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 5 T + 137 T^{2} + 780 T^{3} + 237 p T^{4} + 51855 T^{5} + 473218 T^{6} + 51855 p T^{7} + 237 p^{3} T^{8} + 780 p^{3} T^{9} + 137 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 20 T + 350 T^{2} + 4032 T^{3} + 41631 T^{4} + 343580 T^{5} + 2583236 T^{6} + 343580 p T^{7} + 41631 p^{2} T^{8} + 4032 p^{3} T^{9} + 350 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 13 T + 157 T^{2} + 1612 T^{3} + 18175 T^{4} + 138747 T^{5} + 1067254 T^{6} + 138747 p T^{7} + 18175 p^{2} T^{8} + 1612 p^{3} T^{9} + 157 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 6 T + 198 T^{2} - 1458 T^{3} + 20327 T^{4} - 145956 T^{5} + 1428820 T^{6} - 145956 p T^{7} + 20327 p^{2} T^{8} - 1458 p^{3} T^{9} + 198 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 12 T + 298 T^{2} + 2476 T^{3} + 38311 T^{4} + 255384 T^{5} + 2976300 T^{6} + 255384 p T^{7} + 38311 p^{2} T^{8} + 2476 p^{3} T^{9} + 298 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 3 T + 71 T^{2} - 42 T^{3} + 2543 T^{4} - 43281 T^{5} - 71022 T^{6} - 43281 p T^{7} + 2543 p^{2} T^{8} - 42 p^{3} T^{9} + 71 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 6 T + 280 T^{2} + 1736 T^{3} + 39711 T^{4} + 219082 T^{5} + 3518896 T^{6} + 219082 p T^{7} + 39711 p^{2} T^{8} + 1736 p^{3} T^{9} + 280 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 8 T + 158 T^{2} + 792 T^{3} + 18143 T^{4} + 89824 T^{5} + 1511524 T^{6} + 89824 p T^{7} + 18143 p^{2} T^{8} + 792 p^{3} T^{9} + 158 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 28 T + 574 T^{2} + 7432 T^{3} + 82047 T^{4} + 712284 T^{5} + 6541124 T^{6} + 712284 p T^{7} + 82047 p^{2} T^{8} + 7432 p^{3} T^{9} + 574 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - T + 209 T^{2} - 1098 T^{3} + 22627 T^{4} - 219757 T^{5} + 1796054 T^{6} - 219757 p T^{7} + 22627 p^{2} T^{8} - 1098 p^{3} T^{9} + 209 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 2 T + 268 T^{2} - 272 T^{3} + 39195 T^{4} - 59310 T^{5} + 4212304 T^{6} - 59310 p T^{7} + 39195 p^{2} T^{8} - 272 p^{3} T^{9} + 268 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 19 T + 379 T^{2} + 4574 T^{3} + 64925 T^{4} + 667743 T^{5} + 7703406 T^{6} + 667743 p T^{7} + 64925 p^{2} T^{8} + 4574 p^{3} T^{9} + 379 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.56404155274345610590322048831, −4.15928442947108900941460167596, −4.04881479631744675797569309686, −4.03023221248455685060339870500, −3.98896297403709427212579600916, −3.90044580469210754609677960139, −3.83799884462215112663881434278, −3.48175446576425603569169206277, −3.37739122129172738833371102611, −3.36904859095092386156902624514, −3.35983990370209954107790982747, −3.27566579199969667624942694277, −3.19238570518348667806014261502, −2.61583409328264152414578932428, −2.58181935885320071328559476407, −2.57410321443692296985997070669, −2.50772161336591562539433861660, −2.35405569262579211159434682402, −2.20257515855689948104019540218, −1.65260274871263232032529303261, −1.52976425665196413502846150862, −1.41639902545129245583535921790, −1.39397360537642927394479364352, −1.37304302066897811172493054293, −1.35741643127967554718438929470, 0, 0, 0, 0, 0, 0, 1.35741643127967554718438929470, 1.37304302066897811172493054293, 1.39397360537642927394479364352, 1.41639902545129245583535921790, 1.52976425665196413502846150862, 1.65260274871263232032529303261, 2.20257515855689948104019540218, 2.35405569262579211159434682402, 2.50772161336591562539433861660, 2.57410321443692296985997070669, 2.58181935885320071328559476407, 2.61583409328264152414578932428, 3.19238570518348667806014261502, 3.27566579199969667624942694277, 3.35983990370209954107790982747, 3.36904859095092386156902624514, 3.37739122129172738833371102611, 3.48175446576425603569169206277, 3.83799884462215112663881434278, 3.90044580469210754609677960139, 3.98896297403709427212579600916, 4.03023221248455685060339870500, 4.04881479631744675797569309686, 4.15928442947108900941460167596, 4.56404155274345610590322048831

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

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