Properties

Label 2-91-1.1-c9-0-40
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  − 2.85·2-s + 143.·3-s − 503.·4-s − 15.3·5-s − 410.·6-s + 2.40e3·7-s + 2.90e3·8-s + 957.·9-s + 43.7·10-s − 2.61e3·11-s − 7.23e4·12-s + 2.85e4·13-s − 6.86e3·14-s − 2.19e3·15-s + 2.49e5·16-s + 8.63e4·17-s − 2.73e3·18-s − 9.04e4·19-s + 7.71e3·20-s + 3.44e5·21-s + 7.48e3·22-s − 8.55e5·23-s + 4.17e5·24-s − 1.95e6·25-s − 8.16e4·26-s − 2.69e6·27-s − 1.20e6·28-s + ⋯
L(s)  = 1  − 0.126·2-s + 1.02·3-s − 0.984·4-s − 0.0109·5-s − 0.129·6-s + 0.377·7-s + 0.250·8-s + 0.0486·9-s + 0.00138·10-s − 0.0539·11-s − 1.00·12-s + 0.277·13-s − 0.0477·14-s − 0.0112·15-s + 0.952·16-s + 0.250·17-s − 0.00614·18-s − 0.159·19-s + 0.0107·20-s + 0.387·21-s + 0.00680·22-s − 0.637·23-s + 0.256·24-s − 0.999·25-s − 0.0350·26-s − 0.974·27-s − 0.371·28-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 \( 1 + 2.85T + 512T^{2} \)
3 \( 1 - 143.T + 1.96e4T^{2} \)
5 \( 1 + 15.3T + 1.95e6T^{2} \)
11 \( 1 + 2.61e3T + 2.35e9T^{2} \)
17 \( 1 - 8.63e4T + 1.18e11T^{2} \)
19 \( 1 + 9.04e4T + 3.22e11T^{2} \)
23 \( 1 + 8.55e5T + 1.80e12T^{2} \)
29 \( 1 - 1.60e6T + 1.45e13T^{2} \)
31 \( 1 + 2.47e6T + 2.64e13T^{2} \)
37 \( 1 - 1.00e6T + 1.29e14T^{2} \)
41 \( 1 + 3.12e7T + 3.27e14T^{2} \)
43 \( 1 - 1.84e7T + 5.02e14T^{2} \)
47 \( 1 - 2.65e6T + 1.11e15T^{2} \)
53 \( 1 + 6.38e7T + 3.29e15T^{2} \)
59 \( 1 + 1.84e7T + 8.66e15T^{2} \)
61 \( 1 + 9.45e7T + 1.16e16T^{2} \)
67 \( 1 + 1.40e8T + 2.72e16T^{2} \)
71 \( 1 + 2.69e8T + 4.58e16T^{2} \)
73 \( 1 + 1.26e8T + 5.88e16T^{2} \)
79 \( 1 + 4.38e8T + 1.19e17T^{2} \)
83 \( 1 + 3.83e7T + 1.86e17T^{2} \)
89 \( 1 + 2.45e8T + 3.50e17T^{2} \)
97 \( 1 - 4.47e8T + 7.60e17T^{2} \)
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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

−11.81124727342705626040288769607, −10.33688516314424318904033893712, −9.290204330860415811535658846614, −8.442855353719400688092170721072, −7.66893215998806216198747782772, −5.76417394412302840900311525122, −4.35981200494578833920988729915, −3.23822369745465087511115883025, −1.67850962437510708685361609238, 0, 1.67850962437510708685361609238, 3.23822369745465087511115883025, 4.35981200494578833920988729915, 5.76417394412302840900311525122, 7.66893215998806216198747782772, 8.442855353719400688092170721072, 9.290204330860415811535658846614, 10.33688516314424318904033893712, 11.81124727342705626040288769607

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