Properties

Label 2-7360-1.1-c1-0-145
Degree $2$
Conductor $7360$
Sign $-1$
Analytic cond. $58.7698$
Root an. cond. $7.66615$
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  + 5-s + 7-s − 3·9-s − 2·11-s + 2·13-s + 3·17-s + 2·19-s + 23-s + 25-s − 7·29-s − 5·31-s + 35-s − 11·37-s + 41-s − 3·45-s − 6·49-s − 11·53-s − 2·55-s + 13·59-s + 8·61-s − 3·63-s + 2·65-s − 5·67-s + 5·71-s + 6·73-s − 2·77-s − 12·79-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 9-s − 0.603·11-s + 0.554·13-s + 0.727·17-s + 0.458·19-s + 0.208·23-s + 1/5·25-s − 1.29·29-s − 0.898·31-s + 0.169·35-s − 1.80·37-s + 0.156·41-s − 0.447·45-s − 6/7·49-s − 1.51·53-s − 0.269·55-s + 1.69·59-s + 1.02·61-s − 0.377·63-s + 0.248·65-s − 0.610·67-s + 0.593·71-s + 0.702·73-s − 0.227·77-s − 1.35·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7360\)    =    \(2^{6} \cdot 5 \cdot 23\)
Sign: $-1$
Analytic conductor: \(58.7698\)
Root analytic conductor: \(7.66615\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{7360} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7360,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
23 \( 1 - T \)
good3 \( 1 + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 13 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−7.59653629758399943262516221184, −6.91143568407048423453135395071, −5.97543117230083726419167760169, −5.42647673030720855877781152709, −5.02637245760001494301036384371, −3.75006783743032921485299913724, −3.18806675388433944305091624836, −2.23093209652595357140500085124, −1.36861505540149021722314561914, 0, 1.36861505540149021722314561914, 2.23093209652595357140500085124, 3.18806675388433944305091624836, 3.75006783743032921485299913724, 5.02637245760001494301036384371, 5.42647673030720855877781152709, 5.97543117230083726419167760169, 6.91143568407048423453135395071, 7.59653629758399943262516221184

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