Properties

Label 2-91035-1.1-c1-0-43
Degree $2$
Conductor $91035$
Sign $-1$
Analytic cond. $726.918$
Root an. cond. $26.9614$
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  + 2-s − 4-s + 5-s + 7-s − 3·8-s + 10-s + 14-s − 16-s + 4·19-s − 20-s − 6·23-s + 25-s − 28-s − 6·29-s − 2·31-s + 5·32-s + 35-s + 2·37-s + 4·38-s − 3·40-s + 6·41-s − 8·43-s − 6·46-s + 49-s + 50-s + 2·53-s − 3·56-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.377·7-s − 1.06·8-s + 0.316·10-s + 0.267·14-s − 1/4·16-s + 0.917·19-s − 0.223·20-s − 1.25·23-s + 1/5·25-s − 0.188·28-s − 1.11·29-s − 0.359·31-s + 0.883·32-s + 0.169·35-s + 0.328·37-s + 0.648·38-s − 0.474·40-s + 0.937·41-s − 1.21·43-s − 0.884·46-s + 1/7·49-s + 0.141·50-s + 0.274·53-s − 0.400·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91035\)    =    \(3^{2} \cdot 5 \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(726.918\)
Root analytic conductor: \(26.9614\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 91035,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
17 \( 1 \)
good2 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−13.99521891573114, −13.75600610977989, −13.07263007879546, −12.83069921217150, −12.20880219578010, −11.68687785579293, −11.35995416364822, −10.64857921164970, −10.04214647387980, −9.577057515460135, −9.260439801488864, −8.546923739328599, −8.154163599478901, −7.486530489873961, −7.006364979706599, −6.148643261610972, −5.798740081002803, −5.404920621851151, −4.711501985966460, −4.326400347565051, −3.580688249875278, −3.214054286142616, −2.343412593043992, −1.773744002049336, −0.8983366158294468, 0, 0.8983366158294468, 1.773744002049336, 2.343412593043992, 3.214054286142616, 3.580688249875278, 4.326400347565051, 4.711501985966460, 5.404920621851151, 5.798740081002803, 6.148643261610972, 7.006364979706599, 7.486530489873961, 8.154163599478901, 8.546923739328599, 9.260439801488864, 9.577057515460135, 10.04214647387980, 10.64857921164970, 11.35995416364822, 11.68687785579293, 12.20880219578010, 12.83069921217150, 13.07263007879546, 13.75600610977989, 13.99521891573114

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