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Properties

Label 2-91-13.12-c7-0-37
Degree $2$
Conductor $91$
Sign $0.508 + 0.860i$
Analytic cond. $28.4270$
Root an. cond. $5.33170$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.74i·2-s + 28.2·3-s + 113.·4-s + 90.6i·5-s − 105. i·6-s + 343i·7-s − 905. i·8-s − 1.39e3·9-s + 339.·10-s − 7.61e3i·11-s + 3.21e3·12-s + (6.82e3 − 4.02e3i)13-s + 1.28e3·14-s + 2.55e3i·15-s + 1.11e4·16-s − 983.·17-s + ⋯
L(s)  = 1  − 0.330i·2-s + 0.603·3-s + 0.890·4-s + 0.324i·5-s − 0.199i·6-s + 0.377i·7-s − 0.625i·8-s − 0.635·9-s + 0.107·10-s − 1.72i·11-s + 0.537·12-s + (0.860 − 0.508i)13-s + 0.125·14-s + 0.195i·15-s + 0.683·16-s − 0.0485·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.508 + 0.860i$
Analytic conductor: \(28.4270\)
Root analytic conductor: \(5.33170\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :7/2),\ 0.508 + 0.860i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.61117 - 1.49016i\)
\(L(\frac12)\) \(\approx\) \(2.61117 - 1.49016i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - 343iT \)
13 \( 1 + (-6.82e3 + 4.02e3i)T \)
good2 \( 1 + 3.74iT - 128T^{2} \)
3 \( 1 - 28.2T + 2.18e3T^{2} \)
5 \( 1 - 90.6iT - 7.81e4T^{2} \)
11 \( 1 + 7.61e3iT - 1.94e7T^{2} \)
17 \( 1 + 983.T + 4.10e8T^{2} \)
19 \( 1 + 2.01e4iT - 8.93e8T^{2} \)
23 \( 1 - 4.08e4T + 3.40e9T^{2} \)
29 \( 1 - 1.64e5T + 1.72e10T^{2} \)
31 \( 1 - 2.53e5iT - 2.75e10T^{2} \)
37 \( 1 + 1.83e5iT - 9.49e10T^{2} \)
41 \( 1 + 1.52e5iT - 1.94e11T^{2} \)
43 \( 1 - 8.46e5T + 2.71e11T^{2} \)
47 \( 1 + 4.01e5iT - 5.06e11T^{2} \)
53 \( 1 + 1.51e6T + 1.17e12T^{2} \)
59 \( 1 + 1.07e6iT - 2.48e12T^{2} \)
61 \( 1 + 1.30e6T + 3.14e12T^{2} \)
67 \( 1 - 6.39e5iT - 6.06e12T^{2} \)
71 \( 1 + 8.30e5iT - 9.09e12T^{2} \)
73 \( 1 - 5.38e5iT - 1.10e13T^{2} \)
79 \( 1 + 6.29e6T + 1.92e13T^{2} \)
83 \( 1 - 8.01e6iT - 2.71e13T^{2} \)
89 \( 1 + 1.85e6iT - 4.42e13T^{2} \)
97 \( 1 - 1.74e7iT - 8.07e13T^{2} \)
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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

−12.41590869901466352933554886672, −11.10670854398134861886558158345, −10.79713617824653443144791184432, −8.987031258552219715721897823492, −8.216955231305277884817761231266, −6.68800610095652236421259074487, −5.64892572309154908371978569215, −3.28284771165358551850708647766, −2.76786809132943616137218264644, −0.935088505696185851571556897806, 1.51129185206861560356752379574, 2.80323879440024097646754291583, 4.46540917333020135544342337697, 6.09391643772900225913670400283, 7.24704725753023524212124244349, 8.227388731976304879754751355292, 9.441117004017206370371958093246, 10.71917827216603872787311803690, 11.80559423051192561261341646173, 12.85676114930767076281812035286

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