Properties

Label 2-91-1.1-c7-0-21
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  + 4.83·2-s + 62.4·3-s − 104.·4-s + 314.·5-s + 302.·6-s − 343·7-s − 1.12e3·8-s + 1.71e3·9-s + 1.52e3·10-s + 8.46e3·11-s − 6.53e3·12-s − 2.19e3·13-s − 1.65e3·14-s + 1.96e4·15-s + 7.95e3·16-s + 1.19e4·17-s + 8.30e3·18-s + 6.88e3·19-s − 3.29e4·20-s − 2.14e4·21-s + 4.09e4·22-s + 1.03e5·23-s − 7.02e4·24-s + 2.08e4·25-s − 1.06e4·26-s − 2.93e4·27-s + 3.58e4·28-s + ⋯
L(s)  = 1  + 0.427·2-s + 1.33·3-s − 0.817·4-s + 1.12·5-s + 0.570·6-s − 0.377·7-s − 0.776·8-s + 0.785·9-s + 0.480·10-s + 1.91·11-s − 1.09·12-s − 0.277·13-s − 0.161·14-s + 1.50·15-s + 0.485·16-s + 0.589·17-s + 0.335·18-s + 0.230·19-s − 0.919·20-s − 0.504·21-s + 0.819·22-s + 1.76·23-s − 1.03·24-s + 0.266·25-s − 0.118·26-s − 0.287·27-s + 0.308·28-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

\(L(4)\) \(\approx\) \(4.048164175\)
\(L(\frac12)\) \(\approx\) \(4.048164175\)
\(L(\frac{9}{2})\) 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 \( 1 - 4.83T + 128T^{2} \)
3 \( 1 - 62.4T + 2.18e3T^{2} \)
5 \( 1 - 314.T + 7.81e4T^{2} \)
11 \( 1 - 8.46e3T + 1.94e7T^{2} \)
17 \( 1 - 1.19e4T + 4.10e8T^{2} \)
19 \( 1 - 6.88e3T + 8.93e8T^{2} \)
23 \( 1 - 1.03e5T + 3.40e9T^{2} \)
29 \( 1 + 1.01e5T + 1.72e10T^{2} \)
31 \( 1 - 2.40e5T + 2.75e10T^{2} \)
37 \( 1 + 4.94e5T + 9.49e10T^{2} \)
41 \( 1 + 5.17e5T + 1.94e11T^{2} \)
43 \( 1 - 6.70e5T + 2.71e11T^{2} \)
47 \( 1 - 3.90e5T + 5.06e11T^{2} \)
53 \( 1 - 3.16e5T + 1.17e12T^{2} \)
59 \( 1 - 2.76e6T + 2.48e12T^{2} \)
61 \( 1 - 5.00e4T + 3.14e12T^{2} \)
67 \( 1 + 7.10e5T + 6.06e12T^{2} \)
71 \( 1 + 4.81e6T + 9.09e12T^{2} \)
73 \( 1 - 3.19e5T + 1.10e13T^{2} \)
79 \( 1 - 3.95e5T + 1.92e13T^{2} \)
83 \( 1 + 5.55e6T + 2.71e13T^{2} \)
89 \( 1 + 1.22e7T + 4.42e13T^{2} \)
97 \( 1 + 5.15e6T + 8.07e13T^{2} \)
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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

−13.13316877682861937189408201262, −11.96770551446884157774090515436, −9.949976302156276655946688008677, −9.248086583327558007054097878381, −8.668569065634061126787506507281, −6.90027680631460133428721412824, −5.54303005740611487736982887441, −3.97322238848742945463085317904, −2.92277440972329670489072586058, −1.31425733908645736905751220784, 1.31425733908645736905751220784, 2.92277440972329670489072586058, 3.97322238848742945463085317904, 5.54303005740611487736982887441, 6.90027680631460133428721412824, 8.668569065634061126787506507281, 9.248086583327558007054097878381, 9.949976302156276655946688008677, 11.96770551446884157774090515436, 13.13316877682861937189408201262

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