Properties

Label 12-39e12-1.1-c1e6-0-1
Degree $12$
Conductor $1.238\times 10^{19}$
Sign $1$
Analytic cond. $3.20950\times 10^{6}$
Root an. cond. $3.48500$
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  − 5·4-s + 7·16-s − 2·17-s + 4·25-s + 4·29-s + 30·43-s + 36·49-s + 34·53-s − 26·61-s + 14·64-s + 10·68-s + 6·79-s − 20·100-s + 4·101-s + 28·103-s + 26·107-s − 48·113-s − 20·116-s + 25·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 5/2·4-s + 7/4·16-s − 0.485·17-s + 4/5·25-s + 0.742·29-s + 4.57·43-s + 36/7·49-s + 4.67·53-s − 3.32·61-s + 7/4·64-s + 1.21·68-s + 0.675·79-s − 2·100-s + 0.398·101-s + 2.75·103-s + 2.51·107-s − 4.51·113-s − 1.85·116-s + 2.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(3.20950\times 10^{6}\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1521} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.892514603\)
\(L(\frac12)\) \(\approx\) \(2.892514603\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + 5 T^{2} + 9 p T^{4} + 41 T^{6} + 9 p^{3} T^{8} + 5 p^{4} T^{10} + p^{6} T^{12} \)
5 \( 1 - 4 T^{2} + 36 T^{4} - 241 T^{6} + 36 p^{2} T^{8} - 4 p^{4} T^{10} + p^{6} T^{12} \)
7 \( 1 - 36 T^{2} + 572 T^{4} - 5165 T^{6} + 572 p^{2} T^{8} - 36 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 25 T^{2} + 485 T^{4} - 5601 T^{6} + 485 p^{2} T^{8} - 25 p^{4} T^{10} + p^{6} T^{12} \)
17 \( ( 1 + T + 35 T^{2} + 47 T^{3} + 35 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
19 \( 1 - 93 T^{2} + 3917 T^{4} - 95369 T^{6} + 3917 p^{2} T^{8} - 93 p^{4} T^{10} + p^{6} T^{12} \)
23 \( ( 1 + 20 T^{2} + 91 T^{3} + 20 p T^{4} + p^{3} T^{6} )^{2} \)
29 \( ( 1 - 2 T + 72 T^{2} - 3 p T^{3} + 72 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 12 T^{2} + 824 T^{4} - 48797 T^{6} + 824 p^{2} T^{8} - 12 p^{4} T^{10} + p^{6} T^{12} \)
37 \( 1 - 56 T^{2} + 4660 T^{4} - 150689 T^{6} + 4660 p^{2} T^{8} - 56 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 - 173 T^{2} + 14457 T^{4} - 740849 T^{6} + 14457 p^{2} T^{8} - 173 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 - 15 T + 176 T^{2} - 1331 T^{3} + 176 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 261 T^{2} + 29285 T^{4} - 1807289 T^{6} + 29285 p^{2} T^{8} - 261 p^{4} T^{10} + p^{6} T^{12} \)
53 \( ( 1 - 17 T + 225 T^{2} - 1761 T^{3} + 225 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 286 T^{2} + 37671 T^{4} - 2853988 T^{6} + 37671 p^{2} T^{8} - 286 p^{4} T^{10} + p^{6} T^{12} \)
61 \( ( 1 + 13 T + 167 T^{2} + 1419 T^{3} + 167 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 189 T^{2} + 11465 T^{4} - 439313 T^{6} + 11465 p^{2} T^{8} - 189 p^{4} T^{10} + p^{6} T^{12} \)
71 \( 1 - 244 T^{2} + 32208 T^{4} - 2788141 T^{6} + 32208 p^{2} T^{8} - 244 p^{4} T^{10} + p^{6} T^{12} \)
73 \( 1 - 132 T^{2} + 19884 T^{4} - 1422313 T^{6} + 19884 p^{2} T^{8} - 132 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 - 3 T + 219 T^{2} - 447 T^{3} + 219 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 436 T^{2} + 83400 T^{4} - 8978917 T^{6} + 83400 p^{2} T^{8} - 436 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 - 333 T^{2} + 57485 T^{4} - 6354113 T^{6} + 57485 p^{2} T^{8} - 333 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 + 5 T^{2} + 8710 T^{4} - 929407 T^{6} + 8710 p^{2} T^{8} + 5 p^{4} T^{10} + p^{6} T^{12} \)
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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

−5.00171811512600499612870916933, −4.65153713391220907124012058918, −4.48569166050070118406325560932, −4.36965224806786746449139638394, −4.36080469016562574588959453430, −4.30613812988014028224377894392, −4.12480374571145881596078109018, −3.96162832640416538757463410137, −3.85093449549676267761320583250, −3.63363874953090742454773412152, −3.53081691537283655566765211294, −3.10880848788406165685559886560, −3.01224409845638604410834341729, −2.72400930474258247374171246681, −2.71658151751720849364412184373, −2.40493117856235023151989027897, −2.17297939242587534594821052711, −2.17104215548601367344215788341, −2.12018199270485525594339372696, −1.37316367862143324126032901249, −1.31306878585220114449510101521, −0.889525709410778723954731107381, −0.74513726900819630907797377474, −0.60581511918470796796317907006, −0.38219061254588575890010683466, 0.38219061254588575890010683466, 0.60581511918470796796317907006, 0.74513726900819630907797377474, 0.889525709410778723954731107381, 1.31306878585220114449510101521, 1.37316367862143324126032901249, 2.12018199270485525594339372696, 2.17104215548601367344215788341, 2.17297939242587534594821052711, 2.40493117856235023151989027897, 2.71658151751720849364412184373, 2.72400930474258247374171246681, 3.01224409845638604410834341729, 3.10880848788406165685559886560, 3.53081691537283655566765211294, 3.63363874953090742454773412152, 3.85093449549676267761320583250, 3.96162832640416538757463410137, 4.12480374571145881596078109018, 4.30613812988014028224377894392, 4.36080469016562574588959453430, 4.36965224806786746449139638394, 4.48569166050070118406325560932, 4.65153713391220907124012058918, 5.00171811512600499612870916933

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

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