Properties

Label 2-91-1.1-c9-0-29
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  + 11.5·2-s − 234.·3-s − 378.·4-s − 380.·5-s − 2.70e3·6-s + 2.40e3·7-s − 1.02e4·8-s + 3.54e4·9-s − 4.38e3·10-s + 9.63e3·11-s + 8.90e4·12-s + 2.85e4·13-s + 2.76e4·14-s + 8.93e4·15-s + 7.55e4·16-s + 4.64e5·17-s + 4.09e5·18-s − 3.46e5·19-s + 1.44e5·20-s − 5.64e5·21-s + 1.11e5·22-s + 1.34e6·23-s + 2.41e6·24-s − 1.80e6·25-s + 3.29e5·26-s − 3.71e6·27-s − 9.09e5·28-s + ⋯
L(s)  = 1  + 0.509·2-s − 1.67·3-s − 0.740·4-s − 0.272·5-s − 0.853·6-s + 0.377·7-s − 0.886·8-s + 1.80·9-s − 0.138·10-s + 0.198·11-s + 1.23·12-s + 0.277·13-s + 0.192·14-s + 0.455·15-s + 0.288·16-s + 1.35·17-s + 0.919·18-s − 0.609·19-s + 0.201·20-s − 0.632·21-s + 0.101·22-s + 0.998·23-s + 1.48·24-s − 0.925·25-s + 0.141·26-s − 1.34·27-s − 0.279·28-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 \( 1 - 11.5T + 512T^{2} \)
3 \( 1 + 234.T + 1.96e4T^{2} \)
5 \( 1 + 380.T + 1.95e6T^{2} \)
11 \( 1 - 9.63e3T + 2.35e9T^{2} \)
17 \( 1 - 4.64e5T + 1.18e11T^{2} \)
19 \( 1 + 3.46e5T + 3.22e11T^{2} \)
23 \( 1 - 1.34e6T + 1.80e12T^{2} \)
29 \( 1 - 3.44e6T + 1.45e13T^{2} \)
31 \( 1 + 5.55e5T + 2.64e13T^{2} \)
37 \( 1 + 2.25e7T + 1.29e14T^{2} \)
41 \( 1 + 1.05e7T + 3.27e14T^{2} \)
43 \( 1 - 1.25e7T + 5.02e14T^{2} \)
47 \( 1 + 5.29e7T + 1.11e15T^{2} \)
53 \( 1 - 2.37e7T + 3.29e15T^{2} \)
59 \( 1 - 1.37e8T + 8.66e15T^{2} \)
61 \( 1 + 1.62e8T + 1.16e16T^{2} \)
67 \( 1 - 1.66e8T + 2.72e16T^{2} \)
71 \( 1 - 1.90e8T + 4.58e16T^{2} \)
73 \( 1 - 2.33e8T + 5.88e16T^{2} \)
79 \( 1 - 2.38e8T + 1.19e17T^{2} \)
83 \( 1 + 5.70e8T + 1.86e17T^{2} \)
89 \( 1 - 3.40e8T + 3.50e17T^{2} \)
97 \( 1 + 3.80e8T + 7.60e17T^{2} \)
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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

−11.94210771507581598918507851286, −10.86909887163271797824651231635, −9.814244993419670830126759678048, −8.305509775437828605520071193185, −6.74830241728664660326676294576, −5.58525352103250068088459041211, −4.87694118966584132701974036874, −3.67138169539231515525109397446, −1.14749479305637644379452475727, 0, 1.14749479305637644379452475727, 3.67138169539231515525109397446, 4.87694118966584132701974036874, 5.58525352103250068088459041211, 6.74830241728664660326676294576, 8.305509775437828605520071193185, 9.814244993419670830126759678048, 10.86909887163271797824651231635, 11.94210771507581598918507851286

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