Properties

Label 2-10304-1.1-c1-0-28
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  + 2·5-s + 7-s − 3·9-s + 4·11-s − 4·13-s − 8·17-s + 2·19-s + 23-s − 25-s − 2·29-s − 6·31-s + 2·35-s + 10·37-s + 6·41-s + 8·43-s − 6·45-s + 6·47-s + 49-s − 2·53-s + 8·55-s − 10·61-s − 3·63-s − 8·65-s − 8·67-s − 12·71-s + 6·73-s + 4·77-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s − 9-s + 1.20·11-s − 1.10·13-s − 1.94·17-s + 0.458·19-s + 0.208·23-s − 1/5·25-s − 0.371·29-s − 1.07·31-s + 0.338·35-s + 1.64·37-s + 0.937·41-s + 1.21·43-s − 0.894·45-s + 0.875·47-s + 1/7·49-s − 0.274·53-s + 1.07·55-s − 1.28·61-s − 0.377·63-s − 0.992·65-s − 0.977·67-s − 1.42·71-s + 0.702·73-s + 0.455·77-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 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−17.00850514116158, −16.50798888816475, −15.73821981654066, −14.87432098477466, −14.63448462633815, −14.08742666722615, −13.49352387560223, −12.98794581094341, −12.14137370366255, −11.67738021672324, −11.01570049166958, −10.64698061959144, −9.532742224784304, −9.181650902979701, −8.956519099267935, −7.844762180064473, −7.324232829808658, −6.451405027177130, −6.001936205036168, −5.337003515106189, −4.516825183872497, −3.936773750069066, −2.673271638792613, −2.300453655748188, −1.327043895722752, 0, 1.327043895722752, 2.300453655748188, 2.673271638792613, 3.936773750069066, 4.516825183872497, 5.337003515106189, 6.001936205036168, 6.451405027177130, 7.324232829808658, 7.844762180064473, 8.956519099267935, 9.181650902979701, 9.532742224784304, 10.64698061959144, 11.01570049166958, 11.67738021672324, 12.14137370366255, 12.98794581094341, 13.49352387560223, 14.08742666722615, 14.63448462633815, 14.87432098477466, 15.73821981654066, 16.50798888816475, 17.00850514116158

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