Properties

Label 2-91-1.1-c7-0-11
Degree $2$
Conductor $91$
Sign $1$
Analytic cond. $28.4270$
Root an. cond. $5.33170$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.08·2-s − 25.2·3-s − 102.·4-s + 250.·5-s − 128.·6-s + 343·7-s − 1.16e3·8-s − 1.55e3·9-s + 1.27e3·10-s + 79.1·11-s + 2.57e3·12-s + 2.19e3·13-s + 1.74e3·14-s − 6.31e3·15-s + 7.13e3·16-s − 1.98e4·17-s − 7.87e3·18-s + 6.41e3·19-s − 2.55e4·20-s − 8.65e3·21-s + 402.·22-s + 1.08e5·23-s + 2.95e4·24-s − 1.54e4·25-s + 1.11e4·26-s + 9.43e4·27-s − 3.50e4·28-s + ⋯
L(s)  = 1  + 0.449·2-s − 0.539·3-s − 0.798·4-s + 0.895·5-s − 0.242·6-s + 0.377·7-s − 0.807·8-s − 0.708·9-s + 0.402·10-s + 0.0179·11-s + 0.430·12-s + 0.277·13-s + 0.169·14-s − 0.483·15-s + 0.435·16-s − 0.980·17-s − 0.318·18-s + 0.214·19-s − 0.714·20-s − 0.203·21-s + 0.00805·22-s + 1.85·23-s + 0.435·24-s − 0.198·25-s + 0.124·26-s + 0.922·27-s − 0.301·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $1$
Analytic conductor: \(28.4270\)
Root analytic conductor: \(5.33170\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(1.785802204\)
\(L(\frac12)\) \(\approx\) \(1.785802204\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - 343T \)
13 \( 1 - 2.19e3T \)
good2 \( 1 - 5.08T + 128T^{2} \)
3 \( 1 + 25.2T + 2.18e3T^{2} \)
5 \( 1 - 250.T + 7.81e4T^{2} \)
11 \( 1 - 79.1T + 1.94e7T^{2} \)
17 \( 1 + 1.98e4T + 4.10e8T^{2} \)
19 \( 1 - 6.41e3T + 8.93e8T^{2} \)
23 \( 1 - 1.08e5T + 3.40e9T^{2} \)
29 \( 1 - 2.49e5T + 1.72e10T^{2} \)
31 \( 1 - 6.08e4T + 2.75e10T^{2} \)
37 \( 1 - 1.82e5T + 9.49e10T^{2} \)
41 \( 1 + 8.99e4T + 1.94e11T^{2} \)
43 \( 1 - 3.46e5T + 2.71e11T^{2} \)
47 \( 1 - 6.31e5T + 5.06e11T^{2} \)
53 \( 1 + 8.61e5T + 1.17e12T^{2} \)
59 \( 1 + 1.57e5T + 2.48e12T^{2} \)
61 \( 1 - 2.00e6T + 3.14e12T^{2} \)
67 \( 1 + 9.99e5T + 6.06e12T^{2} \)
71 \( 1 + 3.15e5T + 9.09e12T^{2} \)
73 \( 1 - 3.40e6T + 1.10e13T^{2} \)
79 \( 1 + 6.46e5T + 1.92e13T^{2} \)
83 \( 1 + 8.03e6T + 2.71e13T^{2} \)
89 \( 1 - 3.26e6T + 4.42e13T^{2} \)
97 \( 1 - 3.88e6T + 8.07e13T^{2} \)
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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

−12.86038959640322929147036088790, −11.67720767433113062535744726771, −10.60837826624719479967937635795, −9.311522267387381990394745833106, −8.452175695944079779350966975176, −6.51725580744031961673914647043, −5.50329678897213519535866187153, −4.58280851150161278552839752666, −2.78861643359035278617693336246, −0.836010255614980589089843058380, 0.836010255614980589089843058380, 2.78861643359035278617693336246, 4.58280851150161278552839752666, 5.50329678897213519535866187153, 6.51725580744031961673914647043, 8.452175695944079779350966975176, 9.311522267387381990394745833106, 10.60837826624719479967937635795, 11.67720767433113062535744726771, 12.86038959640322929147036088790

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