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Properties

Label 2-97020-1.1-c1-0-24
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 − 3·13-s + 2·17-s + 2·19-s − 2·23-s + 25-s + 9·31-s + 10·37-s + 12·41-s − 43-s − 10·47-s + 14·53-s − 55-s − 9·59-s − 14·61-s + 3·65-s + 10·67-s + 5·71-s + 11·73-s + 10·79-s + 9·83-s − 2·85-s − 89-s − 2·95-s − 6·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 0.832·13-s + 0.485·17-s + 0.458·19-s − 0.417·23-s + 1/5·25-s + 1.61·31-s + 1.64·37-s + 1.87·41-s − 0.152·43-s − 1.45·47-s + 1.92·53-s − 0.134·55-s − 1.17·59-s − 1.79·61-s + 0.372·65-s + 1.22·67-s + 0.593·71-s + 1.28·73-s + 1.12·79-s + 0.987·83-s − 0.216·85-s − 0.105·89-s − 0.205·95-s − 0.609·97-s + 0.0995·101-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 97020,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.545646850\)
\(L(\frac12)\) \(\approx\) \(2.545646850\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.90542081890586, −13.31884040651176, −12.67817800316051, −12.26130668217537, −11.93679927407357, −11.29288962106297, −11.02142146406034, −10.17901625908874, −9.871478735911547, −9.379439420819376, −8.866040150539856, −8.102184608179015, −7.746216856299807, −7.451833579649175, −6.569045016643164, −6.273197238964313, −5.616116920351327, −4.878900699541370, −4.543338098560337, −3.893302436994915, −3.264618458278880, −2.639354079058628, −2.096109814530243, −1.062961920708135, −0.5912161975841336, 0.5912161975841336, 1.062961920708135, 2.096109814530243, 2.639354079058628, 3.264618458278880, 3.893302436994915, 4.543338098560337, 4.878900699541370, 5.616116920351327, 6.273197238964313, 6.569045016643164, 7.451833579649175, 7.746216856299807, 8.102184608179015, 8.866040150539856, 9.379439420819376, 9.871478735911547, 10.17901625908874, 11.02142146406034, 11.29288962106297, 11.93679927407357, 12.26130668217537, 12.67817800316051, 13.31884040651176, 13.90542081890586

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