Properties

Label 2-91-1.1-c1-0-3
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  + 0.470·2-s + 2.24·3-s − 1.77·4-s + 0.529·5-s + 1.05·6-s − 7-s − 1.77·8-s + 2.05·9-s + 0.249·10-s − 2.24·11-s − 4.00·12-s + 13-s − 0.470·14-s + 1.19·15-s + 2.71·16-s − 1.30·17-s + 0.968·18-s − 1.47·19-s − 0.941·20-s − 2.24·21-s − 1.05·22-s + 5.83·23-s − 4.00·24-s − 4.71·25-s + 0.470·26-s − 2.11·27-s + 1.77·28-s + ⋯
L(s)  = 1  + 0.332·2-s + 1.29·3-s − 0.889·4-s + 0.236·5-s + 0.432·6-s − 0.377·7-s − 0.628·8-s + 0.686·9-s + 0.0787·10-s − 0.678·11-s − 1.15·12-s + 0.277·13-s − 0.125·14-s + 0.307·15-s + 0.679·16-s − 0.317·17-s + 0.228·18-s − 0.337·19-s − 0.210·20-s − 0.490·21-s − 0.225·22-s + 1.21·23-s − 0.816·24-s − 0.943·25-s + 0.0923·26-s − 0.407·27-s + 0.336·28-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

\(L(1)\) \(\approx\) \(1.315528665\)
\(L(\frac12)\) \(\approx\) \(1.315528665\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.470T + 2T^{2} \)
3 \( 1 - 2.24T + 3T^{2} \)
5 \( 1 - 0.529T + 5T^{2} \)
11 \( 1 + 2.24T + 11T^{2} \)
17 \( 1 + 1.30T + 17T^{2} \)
19 \( 1 + 1.47T + 19T^{2} \)
23 \( 1 - 5.83T + 23T^{2} \)
29 \( 1 - 5.22T + 29T^{2} \)
31 \( 1 + 7.02T + 31T^{2} \)
37 \( 1 + 2.36T + 37T^{2} \)
41 \( 1 - 6.49T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 - 8.58T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 15.9T + 67T^{2} \)
71 \( 1 - 1.19T + 71T^{2} \)
73 \( 1 - 7.64T + 73T^{2} \)
79 \( 1 + 1.33T + 79T^{2} \)
83 \( 1 + 16.3T + 83T^{2} \)
89 \( 1 - 6.91T + 89T^{2} \)
97 \( 1 + 3.47T + 97T^{2} \)
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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

−13.92878191226870637036343378418, −13.34452425960702938978586659749, −12.48136097530947487315393465072, −10.64055173704041571584458917132, −9.345127097157821584327934150744, −8.764766573874037691399016881034, −7.55210189783927525293762275792, −5.71590372024041195022493422624, −4.12066848872037228013527344962, −2.79185347271348457784021644014, 2.79185347271348457784021644014, 4.12066848872037228013527344962, 5.71590372024041195022493422624, 7.55210189783927525293762275792, 8.764766573874037691399016881034, 9.345127097157821584327934150744, 10.64055173704041571584458917132, 12.48136097530947487315393465072, 13.34452425960702938978586659749, 13.92878191226870637036343378418

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