Properties

Label 2-102960-1.1-c1-0-15
Degree $2$
Conductor $102960$
Sign $1$
Analytic cond. $822.139$
Root an. cond. $28.6729$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·7-s + 11-s − 13-s + 6·17-s − 8·19-s − 8·23-s + 25-s − 4·29-s + 2·31-s + 2·35-s − 8·37-s + 6·41-s − 12·43-s + 8·47-s − 3·49-s − 12·53-s + 55-s + 8·59-s + 6·61-s − 65-s + 12·67-s + 8·71-s + 2·73-s + 2·77-s + 8·79-s + 12·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s + 0.301·11-s − 0.277·13-s + 1.45·17-s − 1.83·19-s − 1.66·23-s + 1/5·25-s − 0.742·29-s + 0.359·31-s + 0.338·35-s − 1.31·37-s + 0.937·41-s − 1.82·43-s + 1.16·47-s − 3/7·49-s − 1.64·53-s + 0.134·55-s + 1.04·59-s + 0.768·61-s − 0.124·65-s + 1.46·67-s + 0.949·71-s + 0.234·73-s + 0.227·77-s + 0.900·79-s + 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102960\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13\)
Sign: $1$
Analytic conductor: \(822.139\)
Root analytic conductor: \(28.6729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 102960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.319820625\)
\(L(\frac12)\) \(\approx\) \(2.319820625\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−13.88681658214657, −13.23735427306222, −12.66137662124560, −12.24918298508170, −11.91286161965297, −11.18373928545977, −10.83403627535119, −10.21354169421551, −9.839612501196271, −9.401127637534945, −8.625736363969273, −8.172577657273952, −7.943099873490665, −7.197305524065002, −6.527116327834943, −6.212569909998951, −5.444946808850647, −5.154244128929725, −4.427709848781718, −3.830354541063374, −3.402190956959455, −2.350988297211355, −1.998539313101561, −1.418891538259011, −0.4636084179166856, 0.4636084179166856, 1.418891538259011, 1.998539313101561, 2.350988297211355, 3.402190956959455, 3.830354541063374, 4.427709848781718, 5.154244128929725, 5.444946808850647, 6.212569909998951, 6.527116327834943, 7.197305524065002, 7.943099873490665, 8.172577657273952, 8.625736363969273, 9.401127637534945, 9.839612501196271, 10.21354169421551, 10.83403627535119, 11.18373928545977, 11.91286161965297, 12.24918298508170, 12.66137662124560, 13.23735427306222, 13.88681658214657

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