Properties

Label 12-1805e6-1.1-c1e6-0-4
Degree $12$
Conductor $3.458\times 10^{19}$
Sign $1$
Analytic cond. $8.96449\times 10^{6}$
Root an. cond. $3.79644$
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 + 2·5-s + 5·9-s + 2·11-s + 15·16-s + 10·20-s + 5·25-s + 12·29-s + 30·31-s + 25·36-s + 12·41-s + 10·44-s + 10·45-s + 20·49-s + 4·55-s − 20·59-s − 2·61-s + 40·64-s − 2·71-s − 24·79-s + 30·80-s + 10·81-s − 36·89-s + 10·99-s + 25·100-s + 14·101-s + 50·109-s + ⋯
L(s)  = 1  + 5/2·4-s + 0.894·5-s + 5/3·9-s + 0.603·11-s + 15/4·16-s + 2.23·20-s + 25-s + 2.22·29-s + 5.38·31-s + 25/6·36-s + 1.87·41-s + 1.50·44-s + 1.49·45-s + 20/7·49-s + 0.539·55-s − 2.60·59-s − 0.256·61-s + 5·64-s − 0.237·71-s − 2.70·79-s + 3.35·80-s + 10/9·81-s − 3.81·89-s + 1.00·99-s + 5/2·100-s + 1.39·101-s + 4.78·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(5^{6} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(8.96449\times 10^{6}\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 5^{6} \cdot 19^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(53.15807142\)
\(L(\frac12)\) \(\approx\) \(53.15807142\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 2 T - T^{2} + 4 T^{3} - p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
19 \( 1 \)
good2 \( 1 - 5 T^{2} + 5 p T^{4} - 15 T^{6} + 5 p^{3} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} \)
3 \( 1 - 5 T^{2} + 5 p T^{4} - 29 T^{6} + 5 p^{3} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} \)
7 \( 1 - 20 T^{2} + 22 p T^{4} - 873 T^{6} + 22 p^{3} T^{8} - 20 p^{4} T^{10} + p^{6} T^{12} \)
11 \( ( 1 - T + 29 T^{2} - 19 T^{3} + 29 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - 47 T^{2} + 1162 T^{4} - 18711 T^{6} + 1162 p^{2} T^{8} - 47 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 67 T^{2} + 2153 T^{4} - 44221 T^{6} + 2153 p^{2} T^{8} - 67 p^{4} T^{10} + p^{6} T^{12} \)
23 \( 1 - 126 T^{2} + 6838 T^{4} - 205573 T^{6} + 6838 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 - 6 T + 60 T^{2} - 321 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 - 15 T + 163 T^{2} - 1027 T^{3} + 163 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 124 T^{2} + 6918 T^{4} - 272997 T^{6} + 6918 p^{2} T^{8} - 124 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 - 6 T + 2 p T^{2} - 489 T^{3} + 2 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 - 131 T^{2} + 8407 T^{4} - 397137 T^{6} + 8407 p^{2} T^{8} - 131 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 68 T^{2} + 6310 T^{4} - 282273 T^{6} + 6310 p^{2} T^{8} - 68 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 219 T^{2} + 23449 T^{4} - 1541557 T^{6} + 23449 p^{2} T^{8} - 219 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 + 10 T + 122 T^{2} + 889 T^{3} + 122 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 + T + 142 T^{2} + 9 T^{3} + 142 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 326 T^{2} + 48823 T^{4} - 4202580 T^{6} + 48823 p^{2} T^{8} - 326 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 + T + 89 T^{2} + 619 T^{3} + 89 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 202 T^{2} + 24354 T^{4} - 2081577 T^{6} + 24354 p^{2} T^{8} - 202 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 12 T + 205 T^{2} + 1448 T^{3} + 205 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 39 T^{2} + 7249 T^{4} - 843277 T^{6} + 7249 p^{2} T^{8} - 39 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 + 18 T + 295 T^{2} + 3132 T^{3} + 295 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 53 T^{2} + 10075 T^{4} - 1543773 T^{6} + 10075 p^{2} T^{8} - 53 p^{4} T^{10} + p^{6} T^{12} \)
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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

−4.82925627757407190155363858853, −4.52942162536712509200174289071, −4.52106541217634180960866776087, −4.47509036754852416689373231080, −4.25324810995822957970619935575, −4.15239412261466882574791839022, −3.91744215362673922122818733793, −3.74107137521567807169881598465, −3.66410933929767787615637388784, −3.32493686466634782664228709364, −2.99371694333522433594829729451, −2.91102862025525828316609228866, −2.76873810768300137109131327225, −2.62503434664485159513402950379, −2.60611378352894736872901410894, −2.53173245670575417876754799765, −2.45573245177319008314493769308, −1.77369981477163595671865182535, −1.61711170371885654179324319260, −1.61551352178226160454637382829, −1.57972610814576505546006099158, −1.11072972647342762775173189881, −0.983614181473396986055024569303, −0.76930858935707500919458815039, −0.63642656528170702943041939064, 0.63642656528170702943041939064, 0.76930858935707500919458815039, 0.983614181473396986055024569303, 1.11072972647342762775173189881, 1.57972610814576505546006099158, 1.61551352178226160454637382829, 1.61711170371885654179324319260, 1.77369981477163595671865182535, 2.45573245177319008314493769308, 2.53173245670575417876754799765, 2.60611378352894736872901410894, 2.62503434664485159513402950379, 2.76873810768300137109131327225, 2.91102862025525828316609228866, 2.99371694333522433594829729451, 3.32493686466634782664228709364, 3.66410933929767787615637388784, 3.74107137521567807169881598465, 3.91744215362673922122818733793, 4.15239412261466882574791839022, 4.25324810995822957970619935575, 4.47509036754852416689373231080, 4.52106541217634180960866776087, 4.52942162536712509200174289071, 4.82925627757407190155363858853

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

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