Properties

Label 2-97020-1.1-c1-0-22
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 + 4·13-s − 2·17-s − 2·19-s + 8·23-s + 25-s + 4·29-s + 4·31-s − 2·37-s − 12·41-s + 8·43-s − 4·47-s + 10·53-s + 55-s + 4·59-s − 6·61-s − 4·65-s + 12·67-s + 16·73-s + 6·79-s − 2·83-s + 2·85-s + 18·89-s + 2·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s + 1.10·13-s − 0.485·17-s − 0.458·19-s + 1.66·23-s + 1/5·25-s + 0.742·29-s + 0.718·31-s − 0.328·37-s − 1.87·41-s + 1.21·43-s − 0.583·47-s + 1.37·53-s + 0.134·55-s + 0.520·59-s − 0.768·61-s − 0.496·65-s + 1.46·67-s + 1.87·73-s + 0.675·79-s − 0.219·83-s + 0.216·85-s + 1.90·89-s + 0.205·95-s + 0.203·97-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: $\chi_{97020} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 97020,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.595994376\)
\(L(\frac12)\) \(\approx\) \(2.595994376\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−13.66986587145670, −13.37283106202408, −12.80750378117518, −12.37730534936206, −11.77129516849756, −11.28152591303471, −10.88128165480587, −10.44700155338234, −9.913146382848898, −9.190276371101752, −8.639580955859101, −8.474059764276821, −7.820601011090606, −7.173037203397457, −6.649278674691716, −6.326110654955379, −5.555969654562731, −4.889235252125059, −4.622862357216666, −3.659113015250067, −3.472852797278404, −2.629917679539284, −2.044433976432526, −1.093816424671608, −0.5883905977552379, 0.5883905977552379, 1.093816424671608, 2.044433976432526, 2.629917679539284, 3.472852797278404, 3.659113015250067, 4.622862357216666, 4.889235252125059, 5.555969654562731, 6.326110654955379, 6.649278674691716, 7.173037203397457, 7.820601011090606, 8.474059764276821, 8.639580955859101, 9.190276371101752, 9.913146382848898, 10.44700155338234, 10.88128165480587, 11.28152591303471, 11.77129516849756, 12.37730534936206, 12.80750378117518, 13.37283106202408, 13.66986587145670

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