Properties

Label 2-3120-1.1-c1-0-36
Degree $2$
Conductor $3120$
Sign $-1$
Analytic cond. $24.9133$
Root an. cond. $4.99132$
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-s − 5-s − 2·7-s + 9-s − 4·11-s − 13-s − 15-s + 8·17-s + 6·19-s − 2·21-s − 6·23-s + 25-s + 27-s − 4·29-s − 4·33-s + 2·35-s − 2·37-s − 39-s − 2·41-s + 4·43-s − 45-s − 3·49-s + 8·51-s − 10·53-s + 4·55-s + 6·57-s − 4·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.258·15-s + 1.94·17-s + 1.37·19-s − 0.436·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s − 0.696·33-s + 0.338·35-s − 0.328·37-s − 0.160·39-s − 0.312·41-s + 0.609·43-s − 0.149·45-s − 3/7·49-s + 1.12·51-s − 1.37·53-s + 0.539·55-s + 0.794·57-s − 0.520·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(24.9133\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3120} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3120,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−8.001879854227249847712710170108, −7.76948487871725340041744632337, −7.09699921362048702400785248351, −5.88823502695794701483978541538, −5.35993903747378895770836082484, −4.28385424811650436847502480563, −3.22048752563716810372968934226, −2.96849062059581912835324119736, −1.52256173769279382883262599647, 0, 1.52256173769279382883262599647, 2.96849062059581912835324119736, 3.22048752563716810372968934226, 4.28385424811650436847502480563, 5.35993903747378895770836082484, 5.88823502695794701483978541538, 7.09699921362048702400785248351, 7.76948487871725340041744632337, 8.001879854227249847712710170108

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