Properties

Label 2-97020-1.1-c1-0-4
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 − 6·17-s + 2·19-s + 9·23-s + 25-s − 9·29-s − 4·31-s + 2·37-s − 3·41-s + 11·43-s + 12·47-s − 55-s − 7·61-s + 4·65-s + 5·67-s + 6·71-s + 2·73-s − 16·79-s − 9·83-s + 6·85-s − 9·89-s − 2·95-s − 4·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 1.10·13-s − 1.45·17-s + 0.458·19-s + 1.87·23-s + 1/5·25-s − 1.67·29-s − 0.718·31-s + 0.328·37-s − 0.468·41-s + 1.67·43-s + 1.75·47-s − 0.134·55-s − 0.896·61-s + 0.496·65-s + 0.610·67-s + 0.712·71-s + 0.234·73-s − 1.80·79-s − 0.987·83-s + 0.650·85-s − 0.953·89-s − 0.205·95-s − 0.406·97-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

\(L(1)\) \(\approx\) \(1.288099267\)
\(L(\frac12)\) \(\approx\) \(1.288099267\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−13.94148191207230, −13.13287345779594, −12.73609899873038, −12.52860433993932, −11.68404024069724, −11.32389534326829, −10.94794738362240, −10.48446392340252, −9.664412270570643, −9.235155903857455, −8.960150678250078, −8.372844245181251, −7.496374751668251, −7.246688964013767, −6.975422882347614, −6.126125753178434, −5.525758334526917, −5.040153415938034, −4.372439579538273, −4.019006813978509, −3.226921815238823, −2.621753674457707, −2.080381081715793, −1.208221502415283, −0.3752349122712595, 0.3752349122712595, 1.208221502415283, 2.080381081715793, 2.621753674457707, 3.226921815238823, 4.019006813978509, 4.372439579538273, 5.040153415938034, 5.525758334526917, 6.126125753178434, 6.975422882347614, 7.246688964013767, 7.496374751668251, 8.372844245181251, 8.960150678250078, 9.235155903857455, 9.664412270570643, 10.48446392340252, 10.94794738362240, 11.32389534326829, 11.68404024069724, 12.52860433993932, 12.73609899873038, 13.13287345779594, 13.94148191207230

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