Properties

Label 2-91035-1.1-c1-0-38
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  − 2·4-s − 5-s + 7-s + 2·11-s − 5·13-s + 4·16-s + 4·19-s + 2·20-s + 5·23-s + 25-s − 2·28-s − 6·29-s + 3·31-s − 35-s + 7·37-s + 3·41-s + 6·43-s − 4·44-s − 47-s + 49-s + 10·52-s − 4·53-s − 2·55-s + 6·59-s − 61-s − 8·64-s + 5·65-s + ⋯
L(s)  = 1  − 4-s − 0.447·5-s + 0.377·7-s + 0.603·11-s − 1.38·13-s + 16-s + 0.917·19-s + 0.447·20-s + 1.04·23-s + 1/5·25-s − 0.377·28-s − 1.11·29-s + 0.538·31-s − 0.169·35-s + 1.15·37-s + 0.468·41-s + 0.914·43-s − 0.603·44-s − 0.145·47-s + 1/7·49-s + 1.38·52-s − 0.549·53-s − 0.269·55-s + 0.781·59-s − 0.128·61-s − 64-s + 0.620·65-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 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
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

−14.25574541042607, −13.55224749081981, −13.19149870049226, −12.50026361869632, −12.29909708606554, −11.65972543571579, −11.17187635779562, −10.73542800895187, −9.891455214911388, −9.521829180464489, −9.294688627622472, −8.612780121324914, −8.011675486867025, −7.624892500744765, −7.152612990052108, −6.530172682252109, −5.715961592998442, −5.157353629991572, −4.893646641897034, −4.058135698446168, −3.902732384890995, −2.934553651906595, −2.501101045564630, −1.396677005960422, −0.8539212809883762, 0, 0.8539212809883762, 1.396677005960422, 2.501101045564630, 2.934553651906595, 3.902732384890995, 4.058135698446168, 4.893646641897034, 5.157353629991572, 5.715961592998442, 6.530172682252109, 7.152612990052108, 7.624892500744765, 8.011675486867025, 8.612780121324914, 9.294688627622472, 9.521829180464489, 9.891455214911388, 10.73542800895187, 11.17187635779562, 11.65972543571579, 12.29909708606554, 12.50026361869632, 13.19149870049226, 13.55224749081981, 14.25574541042607

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