Properties

Label 2-10304-1.1-c1-0-26
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  − 3·3-s + 4·5-s + 7-s + 6·9-s + 2·11-s − 5·13-s − 12·15-s − 4·19-s − 3·21-s − 23-s + 11·25-s − 9·27-s + 3·29-s − 5·31-s − 6·33-s + 4·35-s − 4·37-s + 15·39-s + 5·41-s − 4·43-s + 24·45-s + 11·47-s + 49-s + 8·55-s + 12·57-s − 12·59-s + 6·61-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.78·5-s + 0.377·7-s + 2·9-s + 0.603·11-s − 1.38·13-s − 3.09·15-s − 0.917·19-s − 0.654·21-s − 0.208·23-s + 11/5·25-s − 1.73·27-s + 0.557·29-s − 0.898·31-s − 1.04·33-s + 0.676·35-s − 0.657·37-s + 2.40·39-s + 0.780·41-s − 0.609·43-s + 3.57·45-s + 1.60·47-s + 1/7·49-s + 1.07·55-s + 1.58·57-s − 1.56·59-s + 0.768·61-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 \( 1 + p T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−17.07292990420764, −16.64120614192065, −16.02166920187525, −15.07786025496834, −14.59503262603858, −13.98144492184856, −13.37202020463035, −12.60835495135929, −12.33554686492731, −11.71875675930458, −10.93176727815936, −10.42879310848341, −10.11394622493159, −9.336384859887165, −8.912242334726688, −7.701423960700302, −6.882948956349970, −6.568759482554845, −5.705484014173152, −5.552453376201101, −4.728807484087240, −4.239205671857218, −2.720927689091296, −1.891223337571582, −1.247293666124359, 0, 1.247293666124359, 1.891223337571582, 2.720927689091296, 4.239205671857218, 4.728807484087240, 5.552453376201101, 5.705484014173152, 6.568759482554845, 6.882948956349970, 7.701423960700302, 8.912242334726688, 9.336384859887165, 10.11394622493159, 10.42879310848341, 10.93176727815936, 11.71875675930458, 12.33554686492731, 12.60835495135929, 13.37202020463035, 13.98144492184856, 14.59503262603858, 15.07786025496834, 16.02166920187525, 16.64120614192065, 17.07292990420764

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