Properties

Label 2-91035-1.1-c1-0-29
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 − 4·11-s − 4·13-s − 14-s − 16-s + 6·19-s + 20-s + 4·22-s + 6·23-s + 25-s + 4·26-s − 28-s + 4·31-s − 5·32-s − 35-s + 4·37-s − 6·38-s − 3·40-s − 2·41-s + 4·43-s + 4·44-s − 6·46-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 − 1.20·11-s − 1.10·13-s − 0.267·14-s − 1/4·16-s + 1.37·19-s + 0.223·20-s + 0.852·22-s + 1.25·23-s + 1/5·25-s + 0.784·26-s − 0.188·28-s + 0.718·31-s − 0.883·32-s − 0.169·35-s + 0.657·37-s − 0.973·38-s − 0.474·40-s − 0.312·41-s + 0.609·43-s + 0.603·44-s − 0.884·46-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.02820493592384, −13.64191595221333, −13.06708823158193, −12.70752067085507, −12.08756420463847, −11.59508219582999, −10.95630025331959, −10.66796334661793, −9.978970803323788, −9.642104277217349, −9.171348270640800, −8.572259612798964, −7.989987360546257, −7.584543231016178, −7.422107829982197, −6.647184996925933, −5.811261541263956, −5.067396775796301, −4.870862878566115, −4.458732001150475, −3.435856171427015, −2.988508518637211, −2.295115752954375, −1.406287032515943, −0.7394045502788123, 0, 0.7394045502788123, 1.406287032515943, 2.295115752954375, 2.988508518637211, 3.435856171427015, 4.458732001150475, 4.870862878566115, 5.067396775796301, 5.811261541263956, 6.647184996925933, 7.422107829982197, 7.584543231016178, 7.989987360546257, 8.572259612798964, 9.171348270640800, 9.642104277217349, 9.978970803323788, 10.66796334661793, 10.95630025331959, 11.59508219582999, 12.08756420463847, 12.70752067085507, 13.06708823158193, 13.64191595221333, 14.02820493592384

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