Properties

Label 2-91-1.1-c9-0-43
Degree $2$
Conductor $91$
Sign $-1$
Analytic cond. $46.8682$
Root an. cond. $6.84603$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 35.9·2-s + 230.·3-s + 780.·4-s + 434.·5-s − 8.29e3·6-s + 2.40e3·7-s − 9.63e3·8-s + 3.35e4·9-s − 1.56e4·10-s − 3.71e4·11-s + 1.80e5·12-s + 2.85e4·13-s − 8.63e4·14-s + 1.00e5·15-s − 5.29e4·16-s − 5.46e5·17-s − 1.20e6·18-s − 9.02e5·19-s + 3.38e5·20-s + 5.53e5·21-s + 1.33e6·22-s − 1.65e6·23-s − 2.22e6·24-s − 1.76e6·25-s − 1.02e6·26-s + 3.19e6·27-s + 1.87e6·28-s + ⋯
L(s)  = 1  − 1.58·2-s + 1.64·3-s + 1.52·4-s + 0.310·5-s − 2.61·6-s + 0.377·7-s − 0.832·8-s + 1.70·9-s − 0.493·10-s − 0.764·11-s + 2.50·12-s + 0.277·13-s − 0.600·14-s + 0.510·15-s − 0.201·16-s − 1.58·17-s − 2.70·18-s − 1.58·19-s + 0.473·20-s + 0.621·21-s + 1.21·22-s − 1.23·23-s − 1.36·24-s − 0.903·25-s − 0.440·26-s + 1.15·27-s + 0.575·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(46.8682\)
Root analytic conductor: \(6.84603\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 91,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - 2.40e3T \)
13 \( 1 - 2.85e4T \)
good2 \( 1 + 35.9T + 512T^{2} \)
3 \( 1 - 230.T + 1.96e4T^{2} \)
5 \( 1 - 434.T + 1.95e6T^{2} \)
11 \( 1 + 3.71e4T + 2.35e9T^{2} \)
17 \( 1 + 5.46e5T + 1.18e11T^{2} \)
19 \( 1 + 9.02e5T + 3.22e11T^{2} \)
23 \( 1 + 1.65e6T + 1.80e12T^{2} \)
29 \( 1 + 2.15e6T + 1.45e13T^{2} \)
31 \( 1 - 2.15e6T + 2.64e13T^{2} \)
37 \( 1 - 1.55e7T + 1.29e14T^{2} \)
41 \( 1 + 2.64e6T + 3.27e14T^{2} \)
43 \( 1 + 3.32e7T + 5.02e14T^{2} \)
47 \( 1 + 4.58e6T + 1.11e15T^{2} \)
53 \( 1 - 1.52e7T + 3.29e15T^{2} \)
59 \( 1 + 1.05e8T + 8.66e15T^{2} \)
61 \( 1 - 5.15e5T + 1.16e16T^{2} \)
67 \( 1 - 2.84e8T + 2.72e16T^{2} \)
71 \( 1 - 3.30e8T + 4.58e16T^{2} \)
73 \( 1 + 1.87e8T + 5.88e16T^{2} \)
79 \( 1 - 4.60e8T + 1.19e17T^{2} \)
83 \( 1 + 1.63e8T + 1.86e17T^{2} \)
89 \( 1 + 8.24e8T + 3.50e17T^{2} \)
97 \( 1 + 6.27e8T + 7.60e17T^{2} \)
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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

−11.19188951692786461005584981953, −10.16636709153216365720692650583, −9.300843090785651537224493360921, −8.350043191944967970539373649116, −7.944647661063523005592762368491, −6.56628734094249382811141880427, −4.20590087945799695810122256995, −2.37335264983263087815734409217, −1.86403527468400059545680605573, 0, 1.86403527468400059545680605573, 2.37335264983263087815734409217, 4.20590087945799695810122256995, 6.56628734094249382811141880427, 7.944647661063523005592762368491, 8.350043191944967970539373649116, 9.300843090785651537224493360921, 10.16636709153216365720692650583, 11.19188951692786461005584981953

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