Properties

Label 2-97020-1.1-c1-0-5
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 11-s − 5·13-s − 2·17-s − 6·19-s − 2·23-s + 25-s + 4·29-s + 9·31-s + 6·37-s + 12·41-s + 43-s − 2·47-s − 6·53-s − 55-s − 59-s + 6·61-s + 5·65-s + 2·67-s − 3·71-s − 3·73-s − 6·79-s + 7·83-s + 2·85-s − 89-s + 6·95-s − 2·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 1.38·13-s − 0.485·17-s − 1.37·19-s − 0.417·23-s + 1/5·25-s + 0.742·29-s + 1.61·31-s + 0.986·37-s + 1.87·41-s + 0.152·43-s − 0.291·47-s − 0.824·53-s − 0.134·55-s − 0.130·59-s + 0.768·61-s + 0.620·65-s + 0.244·67-s − 0.356·71-s − 0.351·73-s − 0.675·79-s + 0.768·83-s + 0.216·85-s − 0.105·89-s + 0.615·95-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: \(0\)
Selberg data: \((2,\ 97020,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.354809095\)
\(L(\frac12)\) \(\approx\) \(1.354809095\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
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

−13.89531804400910, −13.11290914296475, −12.84563821567143, −12.25087979137978, −11.92045580232605, −11.36572379112311, −10.85718328572693, −10.34395213979685, −9.798255338976690, −9.410633800854633, −8.735555865603809, −8.260948221520860, −7.794667259393180, −7.283394419967760, −6.611555985062021, −6.292832912965111, −5.666535618128327, −4.762796817431895, −4.481209270454438, −4.103508637855360, −3.193155876863188, −2.501326688127945, −2.224801039639480, −1.157248770381432, −0.3927951890436051, 0.3927951890436051, 1.157248770381432, 2.224801039639480, 2.501326688127945, 3.193155876863188, 4.103508637855360, 4.481209270454438, 4.762796817431895, 5.666535618128327, 6.292832912965111, 6.611555985062021, 7.283394419967760, 7.794667259393180, 8.260948221520860, 8.735555865603809, 9.410633800854633, 9.798255338976690, 10.34395213979685, 10.85718328572693, 11.36572379112311, 11.92045580232605, 12.25087979137978, 12.84563821567143, 13.11290914296475, 13.89531804400910

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