Properties

Label 2-91-1.1-c1-0-0
Degree $2$
Conductor $91$
Sign $1$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.81·2-s − 3.10·3-s + 1.28·4-s + 2.81·5-s + 5.62·6-s − 7-s + 1.28·8-s + 6.62·9-s − 5.10·10-s + 3.10·11-s − 3.99·12-s + 13-s + 1.81·14-s − 8.72·15-s − 4.91·16-s − 0.524·17-s − 12.0·18-s + 0.813·19-s + 3.62·20-s + 3.10·21-s − 5.62·22-s + 7.33·23-s − 4.00·24-s + 2.91·25-s − 1.81·26-s − 11.2·27-s − 1.28·28-s + ⋯
L(s)  = 1  − 1.28·2-s − 1.79·3-s + 0.644·4-s + 1.25·5-s + 2.29·6-s − 0.377·7-s + 0.455·8-s + 2.20·9-s − 1.61·10-s + 0.935·11-s − 1.15·12-s + 0.277·13-s + 0.484·14-s − 2.25·15-s − 1.22·16-s − 0.127·17-s − 2.83·18-s + 0.186·19-s + 0.811·20-s + 0.677·21-s − 1.19·22-s + 1.53·23-s − 0.816·24-s + 0.583·25-s − 0.355·26-s − 2.16·27-s − 0.243·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $1$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3881012228\)
\(L(\frac12)\) \(\approx\) \(0.3881012228\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T \)
13 \( 1 - T \)
good2 \( 1 + 1.81T + 2T^{2} \)
3 \( 1 + 3.10T + 3T^{2} \)
5 \( 1 - 2.81T + 5T^{2} \)
11 \( 1 - 3.10T + 11T^{2} \)
17 \( 1 + 0.524T + 17T^{2} \)
19 \( 1 - 0.813T + 19T^{2} \)
23 \( 1 - 7.33T + 23T^{2} \)
29 \( 1 - 8.28T + 29T^{2} \)
31 \( 1 - 1.39T + 31T^{2} \)
37 \( 1 + 6.15T + 37T^{2} \)
41 \( 1 + 4.20T + 41T^{2} \)
43 \( 1 - 6.75T + 43T^{2} \)
47 \( 1 + 5.97T + 47T^{2} \)
53 \( 1 + 2.49T + 53T^{2} \)
59 \( 1 + 4.47T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 + 8.72T + 71T^{2} \)
73 \( 1 + 2.34T + 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 - 16.4T + 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + 1.18T + 97T^{2} \)
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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

−13.88777830242513739692539289281, −12.80973583034402151499416733291, −11.59086863743711495577045499367, −10.60099224119543009400133905705, −9.895812578457702131950705188115, −8.948365161903600243579050520670, −6.96372729183181824285675067939, −6.22024670683491927018005029540, −4.88149030753689217866982759788, −1.24925895028719379559500876429, 1.24925895028719379559500876429, 4.88149030753689217866982759788, 6.22024670683491927018005029540, 6.96372729183181824285675067939, 8.948365161903600243579050520670, 9.895812578457702131950705188115, 10.60099224119543009400133905705, 11.59086863743711495577045499367, 12.80973583034402151499416733291, 13.88777830242513739692539289281

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