Properties

Label 2-97020-1.1-c1-0-15
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 + 2·17-s + 6·19-s + 25-s − 2·29-s − 4·31-s − 2·37-s + 4·43-s + 8·47-s + 4·53-s + 55-s − 14·59-s − 14·61-s − 2·65-s − 2·67-s + 8·71-s + 10·73-s − 16·79-s + 4·83-s − 2·85-s + 10·89-s − 6·95-s − 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s + 0.554·13-s + 0.485·17-s + 1.37·19-s + 1/5·25-s − 0.371·29-s − 0.718·31-s − 0.328·37-s + 0.609·43-s + 1.16·47-s + 0.549·53-s + 0.134·55-s − 1.82·59-s − 1.79·61-s − 0.248·65-s − 0.244·67-s + 0.949·71-s + 1.17·73-s − 1.80·79-s + 0.439·83-s − 0.216·85-s + 1.05·89-s − 0.615·95-s − 0.203·97-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

\(L(1)\) \(\approx\) \(2.093643729\)
\(L(\frac12)\) \(\approx\) \(2.093643729\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.76135262221837, −13.44500092518297, −12.68651796685972, −12.27910834884739, −11.93885279761544, −11.27509727000156, −10.84494253518651, −10.48675165771400, −9.767572430132700, −9.245952907365343, −8.946159696804478, −8.135758886099568, −7.775643623235478, −7.306990992845780, −6.834828085829528, −5.980978233432660, −5.662706462085667, −5.078373701749997, −4.429410275140154, −3.822395459085553, −3.259959771472439, −2.814910183174415, −1.892738774476849, −1.226551622149558, −0.4910528494430213, 0.4910528494430213, 1.226551622149558, 1.892738774476849, 2.814910183174415, 3.259959771472439, 3.822395459085553, 4.429410275140154, 5.078373701749997, 5.662706462085667, 5.980978233432660, 6.834828085829528, 7.306990992845780, 7.775643623235478, 8.135758886099568, 8.946159696804478, 9.245952907365343, 9.767572430132700, 10.48675165771400, 10.84494253518651, 11.27509727000156, 11.93885279761544, 12.27910834884739, 12.68651796685972, 13.44500092518297, 13.76135262221837

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