Properties

Label 2-97020-1.1-c1-0-7
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 − 2·13-s − 3·17-s + 7·19-s + 3·23-s + 25-s + 3·29-s + 4·31-s − 10·37-s − 7·43-s − 9·53-s + 55-s + 3·59-s + 61-s + 2·65-s + 2·67-s + 12·71-s + 4·73-s + 2·79-s − 3·83-s + 3·85-s + 9·89-s − 7·95-s + 19·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s − 0.554·13-s − 0.727·17-s + 1.60·19-s + 0.625·23-s + 1/5·25-s + 0.557·29-s + 0.718·31-s − 1.64·37-s − 1.06·43-s − 1.23·53-s + 0.134·55-s + 0.390·59-s + 0.128·61-s + 0.248·65-s + 0.244·67-s + 1.42·71-s + 0.468·73-s + 0.225·79-s − 0.329·83-s + 0.325·85-s + 0.953·89-s − 0.718·95-s + 1.92·97-s + 0.0995·101-s + 0.0985·103-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

\(L(1)\) \(\approx\) \(1.637164879\)
\(L(\frac12)\) \(\approx\) \(1.637164879\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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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

−13.79825968286359, −13.30528893872252, −12.84264648359789, −12.15489236463027, −11.91654396190016, −11.39818015754927, −10.85333085468666, −10.35827733736548, −9.799712560595543, −9.388514974711134, −8.734161925430276, −8.326388755979009, −7.663825223152323, −7.348238966111674, −6.622028497016732, −6.397683283155177, −5.253188666088501, −5.157743075097490, −4.601816730669488, −3.763545565355545, −3.276016508626051, −2.727256344920190, −2.003459483946154, −1.196980711325670, −0.4310786534582770, 0.4310786534582770, 1.196980711325670, 2.003459483946154, 2.727256344920190, 3.276016508626051, 3.763545565355545, 4.601816730669488, 5.157743075097490, 5.253188666088501, 6.397683283155177, 6.622028497016732, 7.348238966111674, 7.663825223152323, 8.326388755979009, 8.734161925430276, 9.388514974711134, 9.799712560595543, 10.35827733736548, 10.85333085468666, 11.39818015754927, 11.91654396190016, 12.15489236463027, 12.84264648359789, 13.30528893872252, 13.79825968286359

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