Properties

Label 2-97020-1.1-c1-0-72
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 11-s + 13-s + 6·17-s + 2·19-s + 7·23-s + 25-s − 5·29-s + 4·31-s − 10·37-s + 9·41-s + 3·43-s − 9·47-s + 9·53-s + 55-s − 12·59-s − 65-s + 8·67-s − 2·71-s − 2·73-s + 6·79-s − 4·83-s − 6·85-s + 6·89-s − 2·95-s − 16·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s + 0.277·13-s + 1.45·17-s + 0.458·19-s + 1.45·23-s + 1/5·25-s − 0.928·29-s + 0.718·31-s − 1.64·37-s + 1.40·41-s + 0.457·43-s − 1.31·47-s + 1.23·53-s + 0.134·55-s − 1.56·59-s − 0.124·65-s + 0.977·67-s − 0.237·71-s − 0.234·73-s + 0.675·79-s − 0.439·83-s − 0.650·85-s + 0.635·89-s − 0.205·95-s − 1.62·97-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: \(1\)
Selberg data: \((2,\ 97020,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−13.98671345340392, −13.61078151143392, −12.95416539081297, −12.57609604153148, −12.08765057960668, −11.61677024333455, −11.04076552582600, −10.67854228311810, −10.11230477767312, −9.536292474384486, −9.101269839755569, −8.522744032937800, −7.944115096919063, −7.537103949421409, −7.079406663912058, −6.468347082782664, −5.782306191459077, −5.265262779249798, −4.899094847739732, −4.069483462143446, −3.520679878769101, −3.056707253579117, −2.434345118947355, −1.419084404659523, −0.9795200026400095, 0, 0.9795200026400095, 1.419084404659523, 2.434345118947355, 3.056707253579117, 3.520679878769101, 4.069483462143446, 4.899094847739732, 5.265262779249798, 5.782306191459077, 6.468347082782664, 7.079406663912058, 7.537103949421409, 7.944115096919063, 8.522744032937800, 9.101269839755569, 9.536292474384486, 10.11230477767312, 10.67854228311810, 11.04076552582600, 11.61677024333455, 12.08765057960668, 12.57609604153148, 12.95416539081297, 13.61078151143392, 13.98671345340392

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