Properties

Label 2-91-1.1-c9-0-22
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  − 0.703·2-s − 141.·3-s − 511.·4-s − 2.24e3·5-s + 99.6·6-s + 2.40e3·7-s + 720.·8-s + 395.·9-s + 1.57e3·10-s + 7.33e4·11-s + 7.24e4·12-s + 2.85e4·13-s − 1.68e3·14-s + 3.18e5·15-s + 2.61e5·16-s − 4.55e5·17-s − 278.·18-s + 1.01e6·19-s + 1.14e6·20-s − 3.40e5·21-s − 5.15e4·22-s − 1.84e6·23-s − 1.02e5·24-s + 3.08e6·25-s − 2.00e4·26-s + 2.73e6·27-s − 1.22e6·28-s + ⋯
L(s)  = 1  − 0.0310·2-s − 1.01·3-s − 0.999·4-s − 1.60·5-s + 0.0314·6-s + 0.377·7-s + 0.0621·8-s + 0.0201·9-s + 0.0499·10-s + 1.50·11-s + 1.00·12-s + 0.277·13-s − 0.0117·14-s + 1.62·15-s + 0.997·16-s − 1.32·17-s − 0.000625·18-s + 1.79·19-s + 1.60·20-s − 0.381·21-s − 0.0469·22-s − 1.37·23-s − 0.0627·24-s + 1.57·25-s − 0.00862·26-s + 0.989·27-s − 0.377·28-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: $\chi_{91} (1, \cdot )$
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 \( 1 + 0.703T + 512T^{2} \)
3 \( 1 + 141.T + 1.96e4T^{2} \)
5 \( 1 + 2.24e3T + 1.95e6T^{2} \)
11 \( 1 - 7.33e4T + 2.35e9T^{2} \)
17 \( 1 + 4.55e5T + 1.18e11T^{2} \)
19 \( 1 - 1.01e6T + 3.22e11T^{2} \)
23 \( 1 + 1.84e6T + 1.80e12T^{2} \)
29 \( 1 + 1.40e6T + 1.45e13T^{2} \)
31 \( 1 + 7.24e6T + 2.64e13T^{2} \)
37 \( 1 - 1.23e7T + 1.29e14T^{2} \)
41 \( 1 - 4.18e6T + 3.27e14T^{2} \)
43 \( 1 - 2.21e7T + 5.02e14T^{2} \)
47 \( 1 + 1.54e7T + 1.11e15T^{2} \)
53 \( 1 - 9.65e7T + 3.29e15T^{2} \)
59 \( 1 + 9.86e7T + 8.66e15T^{2} \)
61 \( 1 - 1.20e8T + 1.16e16T^{2} \)
67 \( 1 + 1.79e8T + 2.72e16T^{2} \)
71 \( 1 - 1.78e8T + 4.58e16T^{2} \)
73 \( 1 - 3.01e8T + 5.88e16T^{2} \)
79 \( 1 + 3.14e8T + 1.19e17T^{2} \)
83 \( 1 + 2.27e8T + 1.86e17T^{2} \)
89 \( 1 + 7.30e8T + 3.50e17T^{2} \)
97 \( 1 - 8.70e7T + 7.60e17T^{2} \)
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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

−11.63982466199503618450682240314, −11.09225872815617637067357770059, −9.392246756927966413080897805819, −8.387156619584199865404786731639, −7.22239752201170860340466064748, −5.74620119906741399013493481912, −4.43800222899875627342762895729, −3.73262071568285528940593985177, −0.979330092345369840804590889079, 0, 0.979330092345369840804590889079, 3.73262071568285528940593985177, 4.43800222899875627342762895729, 5.74620119906741399013493481912, 7.22239752201170860340466064748, 8.387156619584199865404786731639, 9.392246756927966413080897805819, 11.09225872815617637067357770059, 11.63982466199503618450682240314

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