Properties

Label 2-10304-1.1-c1-0-9
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 − 2·5-s − 7-s + 6·9-s − 2·11-s + 13-s + 6·15-s − 2·19-s + 3·21-s − 23-s − 25-s − 9·27-s + 3·29-s − 31-s + 6·33-s + 2·35-s + 2·37-s − 3·39-s − 41-s − 8·43-s − 12·45-s − 5·47-s + 49-s + 6·53-s + 4·55-s + 6·57-s − 6·61-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.894·5-s − 0.377·7-s + 2·9-s − 0.603·11-s + 0.277·13-s + 1.54·15-s − 0.458·19-s + 0.654·21-s − 0.208·23-s − 1/5·25-s − 1.73·27-s + 0.557·29-s − 0.179·31-s + 1.04·33-s + 0.338·35-s + 0.328·37-s − 0.480·39-s − 0.156·41-s − 1.21·43-s − 1.78·45-s − 0.729·47-s + 1/7·49-s + 0.824·53-s + 0.539·55-s + 0.794·57-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
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

−16.71823929113484, −16.46773477067813, −15.75760312174400, −15.48762051789905, −14.87783792222612, −13.84810852758425, −13.25905541547754, −12.54349239080273, −12.29228291817810, −11.53785722614336, −11.22040481528772, −10.60100276634978, −10.06300726267025, −9.448329549338224, −8.332224154362768, −7.954277216453646, −6.978959563562164, −6.673828104786311, −5.884150979176948, −5.322939054848670, −4.615752673969737, −4.010381412639468, −3.191989842731770, −1.954222676505608, −0.7709426696341552, 0, 0.7709426696341552, 1.954222676505608, 3.191989842731770, 4.010381412639468, 4.615752673969737, 5.322939054848670, 5.884150979176948, 6.673828104786311, 6.978959563562164, 7.954277216453646, 8.332224154362768, 9.448329549338224, 10.06300726267025, 10.60100276634978, 11.22040481528772, 11.53785722614336, 12.29228291817810, 12.54349239080273, 13.25905541547754, 13.84810852758425, 14.87783792222612, 15.48762051789905, 15.75760312174400, 16.46773477067813, 16.71823929113484

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