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Properties

Label 2-310464-1.1-c1-0-103
Degree $2$
Conductor $310464$
Sign $1$
Analytic cond. $2479.06$
Root an. cond. $49.7902$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 11-s − 13-s − 2·17-s + 19-s + 4·23-s − 4·25-s − 5·29-s − 4·31-s + 3·37-s + 6·41-s + 2·43-s + 9·47-s + 2·53-s − 55-s − 59-s − 2·61-s + 65-s − 11·67-s − 2·71-s + 11·73-s + 14·79-s + 6·83-s + 2·85-s − 14·89-s − 95-s − 10·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 0.277·13-s − 0.485·17-s + 0.229·19-s + 0.834·23-s − 4/5·25-s − 0.928·29-s − 0.718·31-s + 0.493·37-s + 0.937·41-s + 0.304·43-s + 1.31·47-s + 0.274·53-s − 0.134·55-s − 0.130·59-s − 0.256·61-s + 0.124·65-s − 1.34·67-s − 0.237·71-s + 1.28·73-s + 1.57·79-s + 0.658·83-s + 0.216·85-s − 1.48·89-s − 0.102·95-s − 1.01·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310464\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(2479.06\)
Root analytic conductor: \(49.7902\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 310464,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.677044723\)
\(L(\frac12)\) \(\approx\) \(1.677044723\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−12.43604794147950, −12.30198243568986, −11.79389202145584, −11.11624572579582, −10.98086926575144, −10.56465490022856, −9.697186710901393, −9.468468352133727, −9.073098899471180, −8.520495721158622, −7.975490010420212, −7.461780267649783, −7.215985804241598, −6.649656607271136, −5.990754809496149, −5.643936200824010, −5.071848185604710, −4.455783119715976, −4.017677966561333, −3.600007977699695, −2.910196012278349, −2.368095399639961, −1.786118998003815, −1.047987651158896, −0.3767025633509169, 0.3767025633509169, 1.047987651158896, 1.786118998003815, 2.368095399639961, 2.910196012278349, 3.600007977699695, 4.017677966561333, 4.455783119715976, 5.071848185604710, 5.643936200824010, 5.990754809496149, 6.649656607271136, 7.215985804241598, 7.461780267649783, 7.975490010420212, 8.520495721158622, 9.073098899471180, 9.468468352133727, 9.697186710901393, 10.56465490022856, 10.98086926575144, 11.11624572579582, 11.79389202145584, 12.30198243568986, 12.43604794147950

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