Properties

Label 12-1155e6-1.1-c1e6-0-1
Degree $12$
Conductor $2.374\times 10^{18}$
Sign $1$
Analytic cond. $615395.$
Root an. cond. $3.03689$
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  + 4·4-s + 2·5-s − 3·9-s + 6·11-s + 4·16-s + 18·19-s + 8·20-s + 25-s + 46·29-s + 8·31-s − 12·36-s − 32·41-s + 24·44-s − 6·45-s − 3·49-s + 12·55-s − 2·59-s − 2·61-s − 4·64-s − 16·71-s + 72·76-s + 32·79-s + 8·80-s + 6·81-s + 26·89-s + 36·95-s − 18·99-s + ⋯
L(s)  = 1  + 2·4-s + 0.894·5-s − 9-s + 1.80·11-s + 16-s + 4.12·19-s + 1.78·20-s + 1/5·25-s + 8.54·29-s + 1.43·31-s − 2·36-s − 4.99·41-s + 3.61·44-s − 0.894·45-s − 3/7·49-s + 1.61·55-s − 0.260·59-s − 0.256·61-s − 1/2·64-s − 1.89·71-s + 8.25·76-s + 3.60·79-s + 0.894·80-s + 2/3·81-s + 2.75·89-s + 3.69·95-s − 1.80·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(615395.\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 5^{6} \cdot 7^{6} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(23.49713026\)
\(L(\frac12)\) \(\approx\) \(23.49713026\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T^{2} )^{3} \)
5 \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( ( 1 + T^{2} )^{3} \)
11 \( ( 1 - T )^{6} \)
good2 \( 1 - p^{2} T^{2} + 3 p^{2} T^{4} - 7 p^{2} T^{6} + 3 p^{4} T^{8} - p^{6} T^{10} + p^{6} T^{12} \)
13 \( 1 - 50 T^{2} + 1243 T^{4} - 19712 T^{6} + 1243 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 83 T^{2} + 3150 T^{4} - 68783 T^{6} + 3150 p^{2} T^{8} - 83 p^{4} T^{10} + p^{6} T^{12} \)
19 \( ( 1 - 9 T + 68 T^{2} - 337 T^{3} + 68 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 111 T^{2} + 5558 T^{4} - 162563 T^{6} + 5558 p^{2} T^{8} - 111 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 - 23 T + 260 T^{2} - 1759 T^{3} + 260 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 - 4 T + 61 T^{2} - 250 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 114 T^{2} + 6267 T^{4} - 249568 T^{6} + 6267 p^{2} T^{8} - 114 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 + 16 T + 193 T^{2} + 1362 T^{3} + 193 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 - 139 T^{2} + 11654 T^{4} - 595707 T^{6} + 11654 p^{2} T^{8} - 139 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 2 p T^{2} + 6263 T^{4} - 289872 T^{6} + 6263 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \)
53 \( 1 - 243 T^{2} + 27470 T^{4} - 1839911 T^{6} + 27470 p^{2} T^{8} - 243 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 + T + 128 T^{2} + 203 T^{3} + 128 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 + T + 106 T^{2} - 71 T^{3} + 106 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 + 58 T^{2} + 205 p T^{4} + 511756 T^{6} + 205 p^{3} T^{8} + 58 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 + 8 T + 219 T^{2} + 1134 T^{3} + 219 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 358 T^{2} + 57215 T^{4} - 5315604 T^{6} + 57215 p^{2} T^{8} - 358 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 - 16 T + 285 T^{2} - 2530 T^{3} + 285 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 431 T^{2} + 82158 T^{4} - 8840003 T^{6} + 82158 p^{2} T^{8} - 431 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 - 13 T + 156 T^{2} - 1833 T^{3} + 156 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 239 T^{2} + 37702 T^{4} - 4451063 T^{6} + 37702 p^{2} T^{8} - 239 p^{4} T^{10} + p^{6} T^{12} \)
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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

−5.10329400530480610491740856563, −5.05353503070043138312815769339, −4.93824679287354887967926618076, −4.81613380021400245053599645262, −4.52344768586642535481788896677, −4.45629131431694985322304359297, −4.43332265711404926463108589245, −3.90663839468339991674376607767, −3.78560684874775231478740119355, −3.39322449384621660198858184592, −3.31972430420178243043930732286, −3.28556152065115910334424377970, −3.04390099210720675542262330983, −3.03128626723015262322479776093, −2.86087897167701832064282874163, −2.54386603915423355011414240412, −2.26017807309029508978206207611, −2.25715316595668248819791991148, −2.24106036446187571263682011711, −1.47085574397744101443805261769, −1.42868309678974794188713625687, −1.26459095171238992658129628780, −1.17332481641008675688335520682, −0.861161670016296560402233979296, −0.55258995261200730074051103139, 0.55258995261200730074051103139, 0.861161670016296560402233979296, 1.17332481641008675688335520682, 1.26459095171238992658129628780, 1.42868309678974794188713625687, 1.47085574397744101443805261769, 2.24106036446187571263682011711, 2.25715316595668248819791991148, 2.26017807309029508978206207611, 2.54386603915423355011414240412, 2.86087897167701832064282874163, 3.03128626723015262322479776093, 3.04390099210720675542262330983, 3.28556152065115910334424377970, 3.31972430420178243043930732286, 3.39322449384621660198858184592, 3.78560684874775231478740119355, 3.90663839468339991674376607767, 4.43332265711404926463108589245, 4.45629131431694985322304359297, 4.52344768586642535481788896677, 4.81613380021400245053599645262, 4.93824679287354887967926618076, 5.05353503070043138312815769339, 5.10329400530480610491740856563

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

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