Properties

Label 2-10304-1.1-c1-0-30
Degree $2$
Conductor $10304$
Sign $-1$
Analytic cond. $82.2778$
Root an. cond. $9.07071$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 7-s + 9-s − 4·11-s − 2·13-s + 4·19-s + 2·21-s − 23-s − 5·25-s − 4·27-s + 6·29-s − 2·31-s − 8·33-s + 2·37-s − 4·39-s − 6·41-s − 4·43-s + 10·47-s + 49-s + 6·53-s + 8·57-s + 6·59-s − 4·61-s + 63-s − 4·67-s − 2·69-s − 8·71-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.917·19-s + 0.436·21-s − 0.208·23-s − 25-s − 0.769·27-s + 1.11·29-s − 0.359·31-s − 1.39·33-s + 0.328·37-s − 0.640·39-s − 0.937·41-s − 0.609·43-s + 1.45·47-s + 1/7·49-s + 0.824·53-s + 1.05·57-s + 0.781·59-s − 0.512·61-s + 0.125·63-s − 0.488·67-s − 0.240·69-s − 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10304\)    =    \(2^{6} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(82.2778\)
Root analytic conductor: \(9.07071\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10304} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 10304,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−16.86014537271703, −16.12721721776572, −15.63973260108531, −15.09334958712454, −14.63100236032693, −13.94011604540266, −13.54897787746538, −13.15086493320586, −12.12907220344678, −11.87536085298761, −10.98518015366856, −10.28496612782706, −9.825830988097532, −9.163512679778035, −8.461216886627752, −7.998123075616248, −7.511216491978507, −6.881784068896763, −5.723590035834086, −5.314412166325533, −4.432085603592244, −3.674197759076136, −2.787924486251786, −2.442679733706256, −1.432793377382518, 0, 1.432793377382518, 2.442679733706256, 2.787924486251786, 3.674197759076136, 4.432085603592244, 5.314412166325533, 5.723590035834086, 6.881784068896763, 7.511216491978507, 7.998123075616248, 8.461216886627752, 9.163512679778035, 9.825830988097532, 10.28496612782706, 10.98518015366856, 11.87536085298761, 12.12907220344678, 13.15086493320586, 13.54897787746538, 13.94011604540266, 14.63100236032693, 15.09334958712454, 15.63973260108531, 16.12721721776572, 16.86014537271703

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