Properties

Label 2-310464-1.1-c1-0-112
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  + 2·5-s − 11-s − 2·13-s + 2·17-s + 4·19-s + 6·23-s − 25-s − 6·29-s − 10·31-s − 2·37-s − 2·41-s − 10·43-s − 6·53-s − 2·55-s + 12·59-s + 2·61-s − 4·65-s + 12·67-s − 2·71-s − 2·73-s + 14·83-s + 4·85-s + 10·89-s + 8·95-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.301·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s + 1.25·23-s − 1/5·25-s − 1.11·29-s − 1.79·31-s − 0.328·37-s − 0.312·41-s − 1.52·43-s − 0.824·53-s − 0.269·55-s + 1.56·59-s + 0.256·61-s − 0.496·65-s + 1.46·67-s − 0.237·71-s − 0.234·73-s + 1.53·83-s + 0.433·85-s + 1.05·89-s + 0.820·95-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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

\(L(1)\) \(\approx\) \(2.360885141\)
\(L(\frac12)\) \(\approx\) \(2.360885141\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−12.76217577009878, −12.25817006640311, −11.69171758956312, −11.26124097644579, −10.88814727730182, −10.24043293874713, −9.862087899530336, −9.544963697384710, −9.057041063911317, −8.637232488809816, −7.957599826269677, −7.427955788480846, −7.191789316062909, −6.562464510016935, −6.049435588485702, −5.369601092675636, −5.208949327033560, −4.867096681363317, −3.773960736614548, −3.554414447643765, −2.932320880862635, −2.216567886011877, −1.841769037609612, −1.207647833191305, −0.4033951067375227, 0.4033951067375227, 1.207647833191305, 1.841769037609612, 2.216567886011877, 2.932320880862635, 3.554414447643765, 3.773960736614548, 4.867096681363317, 5.208949327033560, 5.369601092675636, 6.049435588485702, 6.562464510016935, 7.191789316062909, 7.427955788480846, 7.957599826269677, 8.637232488809816, 9.057041063911317, 9.544963697384710, 9.862087899530336, 10.24043293874713, 10.88814727730182, 11.26124097644579, 11.69171758956312, 12.25817006640311, 12.76217577009878

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