Properties

Label 2-97020-1.1-c1-0-45
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 − 4·13-s − 3·17-s + 5·19-s − 3·23-s + 25-s + 3·29-s − 4·31-s + 11·37-s − 6·41-s − 43-s − 3·47-s − 12·53-s − 55-s + 15·59-s − 10·61-s + 4·65-s − 4·67-s + 9·71-s + 2·73-s − 16·79-s + 12·83-s + 3·85-s − 12·89-s − 5·95-s + 17·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 1.10·13-s − 0.727·17-s + 1.14·19-s − 0.625·23-s + 1/5·25-s + 0.557·29-s − 0.718·31-s + 1.80·37-s − 0.937·41-s − 0.152·43-s − 0.437·47-s − 1.64·53-s − 0.134·55-s + 1.95·59-s − 1.28·61-s + 0.496·65-s − 0.488·67-s + 1.06·71-s + 0.234·73-s − 1.80·79-s + 1.31·83-s + 0.325·85-s − 1.27·89-s − 0.512·95-s + 1.72·97-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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−14.20071655124558, −13.46373694817781, −13.07611781731513, −12.52185220202954, −11.99269495608889, −11.63524420665220, −11.20994277154166, −10.62692337278228, −9.928742805768039, −9.652817252458557, −9.140348694936005, −8.517163953296691, −7.893125888482112, −7.618137722683139, −6.930363331753076, −6.566748305081181, −5.847519709828756, −5.278944727769971, −4.630674091001532, −4.328827656099670, −3.500549156710310, −3.018854142749276, −2.322082932265167, −1.667468792420814, −0.7852024296666054, 0, 0.7852024296666054, 1.667468792420814, 2.322082932265167, 3.018854142749276, 3.500549156710310, 4.328827656099670, 4.630674091001532, 5.278944727769971, 5.847519709828756, 6.566748305081181, 6.930363331753076, 7.618137722683139, 7.893125888482112, 8.517163953296691, 9.140348694936005, 9.652817252458557, 9.928742805768039, 10.62692337278228, 11.20994277154166, 11.63524420665220, 11.99269495608889, 12.52185220202954, 13.07611781731513, 13.46373694817781, 14.20071655124558

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