Properties

Degree $2$
Conductor $302016$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 9-s + 13-s − 2·15-s + 6·17-s − 4·19-s − 8·23-s − 25-s − 27-s − 10·29-s − 6·37-s − 39-s − 10·41-s + 4·43-s + 2·45-s + 8·47-s − 7·49-s − 6·51-s + 10·53-s + 4·57-s + 12·59-s + 14·61-s + 2·65-s + 12·67-s + 8·69-s + 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.277·13-s − 0.516·15-s + 1.45·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.986·37-s − 0.160·39-s − 1.56·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s − 49-s − 0.840·51-s + 1.37·53-s + 0.529·57-s + 1.56·59-s + 1.79·61-s + 0.248·65-s + 1.46·67-s + 0.963·69-s + 0.702·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 302016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 302016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(302016\)    =    \(2^{6} \cdot 3 \cdot 11^{2} \cdot 13\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{302016} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 302016,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
11 \( 1 \)
13 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.88751613807893, −12.50851416159660, −11.82122342122240, −11.74017674663030, −11.05540363622918, −10.52020400586110, −10.13427344237055, −9.820010470933045, −9.448610709931309, −8.699076082549111, −8.325356817250144, −7.828243345161674, −7.210021653595910, −6.809332344544512, −6.200081957835601, −5.783577857000459, −5.415288381577162, −5.137872428982507, −4.188156639925646, −3.724781485577871, −3.487728786244065, −2.243906760659086, −2.171587402866433, −1.501051073653332, −0.7593096177862281, 0, 0.7593096177862281, 1.501051073653332, 2.171587402866433, 2.243906760659086, 3.487728786244065, 3.724781485577871, 4.188156639925646, 5.137872428982507, 5.415288381577162, 5.783577857000459, 6.200081957835601, 6.809332344544512, 7.210021653595910, 7.828243345161674, 8.325356817250144, 8.699076082549111, 9.448610709931309, 9.820010470933045, 10.13427344237055, 10.52020400586110, 11.05540363622918, 11.74017674663030, 11.82122342122240, 12.50851416159660, 12.88751613807893

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