Properties

Label 2-97020-1.1-c1-0-81
Degree $2$
Conductor $97020$
Sign $1$
Analytic cond. $774.708$
Root an. cond. $27.8335$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 11-s − 5·13-s − 2·17-s − 19-s − 2·23-s + 25-s − 6·29-s − 31-s + 37-s − 8·41-s + 43-s − 2·47-s + 4·53-s − 55-s − 6·59-s − 14·61-s + 5·65-s − 13·67-s − 8·71-s − 13·73-s − 11·79-s − 8·83-s + 2·85-s + 14·89-s + 95-s − 2·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 1.38·13-s − 0.485·17-s − 0.229·19-s − 0.417·23-s + 1/5·25-s − 1.11·29-s − 0.179·31-s + 0.164·37-s − 1.24·41-s + 0.152·43-s − 0.291·47-s + 0.549·53-s − 0.134·55-s − 0.781·59-s − 1.79·61-s + 0.620·65-s − 1.58·67-s − 0.949·71-s − 1.52·73-s − 1.23·79-s − 0.878·83-s + 0.216·85-s + 1.48·89-s + 0.102·95-s − 0.203·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97020\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(774.708\)
Root analytic conductor: \(27.8335\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{97020} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 97020,\ (\ :1/2),\ 1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 - T \)
good13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.47016718566668, −13.66749922702040, −13.35909558178726, −12.75819453614966, −12.24712924401522, −11.82025174640714, −11.51069406446654, −10.77724206946614, −10.35691118769342, −9.886316695940639, −9.120471348710054, −9.033313738448538, −8.214917447023968, −7.702746942271872, −7.246601893410082, −6.856597931672489, −6.097272374098543, −5.652834956215260, −4.906545191787095, −4.458568922155744, −3.995823731152628, −3.186331817764054, −2.726082066853515, −1.915294309863168, −1.388784666975925, 0, 0, 1.388784666975925, 1.915294309863168, 2.726082066853515, 3.186331817764054, 3.995823731152628, 4.458568922155744, 4.906545191787095, 5.652834956215260, 6.097272374098543, 6.856597931672489, 7.246601893410082, 7.702746942271872, 8.214917447023968, 9.033313738448538, 9.120471348710054, 9.886316695940639, 10.35691118769342, 10.77724206946614, 11.51069406446654, 11.82025174640714, 12.24712924401522, 12.75819453614966, 13.35909558178726, 13.66749922702040, 14.47016718566668

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