Properties

Label 2-91-1.1-c9-0-46
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  + 29.8·2-s − 70.1·3-s + 381.·4-s + 1.11e3·5-s − 2.09e3·6-s + 2.40e3·7-s − 3.89e3·8-s − 1.47e4·9-s + 3.32e4·10-s − 3.34e4·11-s − 2.67e4·12-s + 2.85e4·13-s + 7.17e4·14-s − 7.81e4·15-s − 3.11e5·16-s − 1.57e5·17-s − 4.41e5·18-s + 3.09e5·19-s + 4.25e5·20-s − 1.68e5·21-s − 1.00e6·22-s − 7.96e5·23-s + 2.73e5·24-s − 7.12e5·25-s + 8.53e5·26-s + 2.41e6·27-s + 9.16e5·28-s + ⋯
L(s)  = 1  + 1.32·2-s − 0.499·3-s + 0.745·4-s + 0.796·5-s − 0.660·6-s + 0.377·7-s − 0.336·8-s − 0.750·9-s + 1.05·10-s − 0.689·11-s − 0.372·12-s + 0.277·13-s + 0.499·14-s − 0.398·15-s − 1.18·16-s − 0.456·17-s − 0.990·18-s + 0.545·19-s + 0.594·20-s − 0.188·21-s − 0.911·22-s − 0.593·23-s + 0.168·24-s − 0.364·25-s + 0.366·26-s + 0.874·27-s + 0.281·28-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 \( 1 - 29.8T + 512T^{2} \)
3 \( 1 + 70.1T + 1.96e4T^{2} \)
5 \( 1 - 1.11e3T + 1.95e6T^{2} \)
11 \( 1 + 3.34e4T + 2.35e9T^{2} \)
17 \( 1 + 1.57e5T + 1.18e11T^{2} \)
19 \( 1 - 3.09e5T + 3.22e11T^{2} \)
23 \( 1 + 7.96e5T + 1.80e12T^{2} \)
29 \( 1 + 1.10e6T + 1.45e13T^{2} \)
31 \( 1 + 2.97e6T + 2.64e13T^{2} \)
37 \( 1 - 4.80e6T + 1.29e14T^{2} \)
41 \( 1 + 2.00e7T + 3.27e14T^{2} \)
43 \( 1 + 3.06e7T + 5.02e14T^{2} \)
47 \( 1 + 6.06e7T + 1.11e15T^{2} \)
53 \( 1 + 4.36e7T + 3.29e15T^{2} \)
59 \( 1 + 3.11e6T + 8.66e15T^{2} \)
61 \( 1 - 3.50e6T + 1.16e16T^{2} \)
67 \( 1 - 2.57e8T + 2.72e16T^{2} \)
71 \( 1 + 1.34e8T + 4.58e16T^{2} \)
73 \( 1 - 2.83e8T + 5.88e16T^{2} \)
79 \( 1 - 1.55e7T + 1.19e17T^{2} \)
83 \( 1 - 4.31e8T + 1.86e17T^{2} \)
89 \( 1 + 5.70e8T + 3.50e17T^{2} \)
97 \( 1 - 2.89e7T + 7.60e17T^{2} \)
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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

−11.84218425476401719833006929868, −11.06345546831847084575418409227, −9.689616027394414620788402895983, −8.296275663040301795601268568107, −6.52949889327216136585002957629, −5.60427916233106354976299748579, −4.90335598732201873427974249121, −3.33734785931127690206794973386, −2.02451089522675186898418455731, 0, 2.02451089522675186898418455731, 3.33734785931127690206794973386, 4.90335598732201873427974249121, 5.60427916233106354976299748579, 6.52949889327216136585002957629, 8.296275663040301795601268568107, 9.689616027394414620788402895983, 11.06345546831847084575418409227, 11.84218425476401719833006929868

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