Properties

Label 2-91-1.1-c9-0-30
Degree $2$
Conductor $91$
Sign $-1$
Analytic cond. $46.8682$
Root an. cond. $6.84603$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 36.7·2-s − 181.·3-s + 837.·4-s + 2.26e3·5-s + 6.65e3·6-s − 2.40e3·7-s − 1.19e4·8-s + 1.31e4·9-s − 8.31e4·10-s + 4.94e3·11-s − 1.51e5·12-s − 2.85e4·13-s + 8.82e4·14-s − 4.10e5·15-s + 1.05e4·16-s − 1.78e5·17-s − 4.82e5·18-s − 4.71e5·19-s + 1.89e6·20-s + 4.34e5·21-s − 1.81e5·22-s + 2.11e6·23-s + 2.16e6·24-s + 3.17e6·25-s + 1.04e6·26-s + 1.18e6·27-s − 2.01e6·28-s + ⋯
L(s)  = 1  − 1.62·2-s − 1.29·3-s + 1.63·4-s + 1.61·5-s + 2.09·6-s − 0.377·7-s − 1.03·8-s + 0.667·9-s − 2.62·10-s + 0.101·11-s − 2.11·12-s − 0.277·13-s + 0.613·14-s − 2.09·15-s + 0.0402·16-s − 0.518·17-s − 1.08·18-s − 0.830·19-s + 2.64·20-s + 0.488·21-s − 0.165·22-s + 1.57·23-s + 1.33·24-s + 1.62·25-s + 0.450·26-s + 0.429·27-s − 0.618·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(46.8682\)
Root analytic conductor: \(6.84603\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 91,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + 2.40e3T \)
13 \( 1 + 2.85e4T \)
good2 \( 1 + 36.7T + 512T^{2} \)
3 \( 1 + 181.T + 1.96e4T^{2} \)
5 \( 1 - 2.26e3T + 1.95e6T^{2} \)
11 \( 1 - 4.94e3T + 2.35e9T^{2} \)
17 \( 1 + 1.78e5T + 1.18e11T^{2} \)
19 \( 1 + 4.71e5T + 3.22e11T^{2} \)
23 \( 1 - 2.11e6T + 1.80e12T^{2} \)
29 \( 1 + 4.05e6T + 1.45e13T^{2} \)
31 \( 1 - 1.98e6T + 2.64e13T^{2} \)
37 \( 1 + 9.92e6T + 1.29e14T^{2} \)
41 \( 1 - 8.06e6T + 3.27e14T^{2} \)
43 \( 1 + 3.04e7T + 5.02e14T^{2} \)
47 \( 1 - 5.28e7T + 1.11e15T^{2} \)
53 \( 1 + 6.65e6T + 3.29e15T^{2} \)
59 \( 1 - 1.63e7T + 8.66e15T^{2} \)
61 \( 1 + 1.19e8T + 1.16e16T^{2} \)
67 \( 1 - 2.03e8T + 2.72e16T^{2} \)
71 \( 1 - 4.61e7T + 4.58e16T^{2} \)
73 \( 1 - 1.31e8T + 5.88e16T^{2} \)
79 \( 1 - 4.76e8T + 1.19e17T^{2} \)
83 \( 1 + 9.05e7T + 1.86e17T^{2} \)
89 \( 1 + 4.07e7T + 3.50e17T^{2} \)
97 \( 1 + 6.02e8T + 7.60e17T^{2} \)
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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

−11.12422839646273493392793305262, −10.52148769617441659936346826404, −9.597570243911068025511495749323, −8.793549405722196507864619046815, −6.97426671869014053725515378457, −6.27099278008808870669793748346, −5.13220400001161868198723621436, −2.34594376063755238405251887039, −1.17708956523138311231487209647, 0, 1.17708956523138311231487209647, 2.34594376063755238405251887039, 5.13220400001161868198723621436, 6.27099278008808870669793748346, 6.97426671869014053725515378457, 8.793549405722196507864619046815, 9.597570243911068025511495749323, 10.52148769617441659936346826404, 11.12422839646273493392793305262

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