Properties

Label 2-91-1.1-c7-0-14
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 22.0·2-s − 47.1·3-s + 358.·4-s − 467.·5-s + 1.03e3·6-s + 343·7-s − 5.07e3·8-s + 32.8·9-s + 1.03e4·10-s − 4.96e3·11-s − 1.68e4·12-s − 2.19e3·13-s − 7.56e3·14-s + 2.20e4·15-s + 6.60e4·16-s + 2.04e4·17-s − 723.·18-s + 1.12e4·19-s − 1.67e5·20-s − 1.61e4·21-s + 1.09e5·22-s + 6.29e4·23-s + 2.39e5·24-s + 1.40e5·25-s + 4.84e4·26-s + 1.01e5·27-s + 1.22e5·28-s + ⋯
L(s)  = 1  − 1.94·2-s − 1.00·3-s + 2.79·4-s − 1.67·5-s + 1.96·6-s + 0.377·7-s − 3.50·8-s + 0.0150·9-s + 3.26·10-s − 1.12·11-s − 2.81·12-s − 0.277·13-s − 0.736·14-s + 1.68·15-s + 4.03·16-s + 1.01·17-s − 0.0292·18-s + 0.376·19-s − 4.68·20-s − 0.380·21-s + 2.19·22-s + 1.07·23-s + 3.53·24-s + 1.79·25-s + 0.540·26-s + 0.992·27-s + 1.05·28-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 91,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 + 22.0T + 128T^{2} \)
3 \( 1 + 47.1T + 2.18e3T^{2} \)
5 \( 1 + 467.T + 7.81e4T^{2} \)
11 \( 1 + 4.96e3T + 1.94e7T^{2} \)
17 \( 1 - 2.04e4T + 4.10e8T^{2} \)
19 \( 1 - 1.12e4T + 8.93e8T^{2} \)
23 \( 1 - 6.29e4T + 3.40e9T^{2} \)
29 \( 1 + 1.55e4T + 1.72e10T^{2} \)
31 \( 1 - 5.08e4T + 2.75e10T^{2} \)
37 \( 1 - 2.68e5T + 9.49e10T^{2} \)
41 \( 1 + 5.24e5T + 1.94e11T^{2} \)
43 \( 1 + 1.02e6T + 2.71e11T^{2} \)
47 \( 1 - 9.37e5T + 5.06e11T^{2} \)
53 \( 1 - 1.65e6T + 1.17e12T^{2} \)
59 \( 1 + 2.20e6T + 2.48e12T^{2} \)
61 \( 1 - 2.30e5T + 3.14e12T^{2} \)
67 \( 1 - 2.00e6T + 6.06e12T^{2} \)
71 \( 1 - 2.93e6T + 9.09e12T^{2} \)
73 \( 1 - 1.33e6T + 1.10e13T^{2} \)
79 \( 1 + 2.92e6T + 1.92e13T^{2} \)
83 \( 1 + 7.49e6T + 2.71e13T^{2} \)
89 \( 1 + 1.80e6T + 4.42e13T^{2} \)
97 \( 1 + 5.26e6T + 8.07e13T^{2} \)
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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

−11.64328765147912623286708316866, −11.01179853670778856673997241070, −10.13068692787760605780609832973, −8.538523286028735884366889808429, −7.80159390069477023775038555195, −6.95865471710102763962326718463, −5.31824118543435624621734096679, −2.99877221436532285126095867145, −0.894887741202150934763324009855, 0, 0.894887741202150934763324009855, 2.99877221436532285126095867145, 5.31824118543435624621734096679, 6.95865471710102763962326718463, 7.80159390069477023775038555195, 8.538523286028735884366889808429, 10.13068692787760605780609832973, 11.01179853670778856673997241070, 11.64328765147912623286708316866

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