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Properties

Label 2-91-13.12-c7-0-3
Degree $2$
Conductor $91$
Sign $-0.00470 - 0.999i$
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  − 15.7i·2-s + 36.7·3-s − 119.·4-s + 452. i·5-s − 578. i·6-s + 343i·7-s − 132. i·8-s − 837.·9-s + 7.11e3·10-s − 5.58e3i·11-s − 4.39e3·12-s + (−7.92e3 + 37.3i)13-s + 5.39e3·14-s + 1.66e4i·15-s − 1.73e4·16-s − 1.70e4·17-s + ⋯
L(s)  = 1  − 1.39i·2-s + 0.785·3-s − 0.934·4-s + 1.61i·5-s − 1.09i·6-s + 0.377i·7-s − 0.0914i·8-s − 0.382·9-s + 2.25·10-s − 1.26i·11-s − 0.733·12-s + (−0.999 + 0.00470i)13-s + 0.525·14-s + 1.27i·15-s − 1.06·16-s − 0.841·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00470 - 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.00470 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.00470 - 0.999i$
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: \((2,\ 91,\ (\ :7/2),\ -0.00470 - 0.999i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.378988 + 0.380777i\)
\(L(\frac12)\) \(\approx\) \(0.378988 + 0.380777i\)
\(L(\frac{9}{2})\) 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 \( 1 + (7.92e3 - 37.3i)T \)
good2 \( 1 + 15.7iT - 128T^{2} \)
3 \( 1 - 36.7T + 2.18e3T^{2} \)
5 \( 1 - 452. iT - 7.81e4T^{2} \)
11 \( 1 + 5.58e3iT - 1.94e7T^{2} \)
17 \( 1 + 1.70e4T + 4.10e8T^{2} \)
19 \( 1 - 5.26e4iT - 8.93e8T^{2} \)
23 \( 1 - 4.05e4T + 3.40e9T^{2} \)
29 \( 1 + 1.74e5T + 1.72e10T^{2} \)
31 \( 1 - 1.47e4iT - 2.75e10T^{2} \)
37 \( 1 - 3.02e4iT - 9.49e10T^{2} \)
41 \( 1 + 6.64e5iT - 1.94e11T^{2} \)
43 \( 1 - 8.18e4T + 2.71e11T^{2} \)
47 \( 1 - 1.31e6iT - 5.06e11T^{2} \)
53 \( 1 + 6.16e5T + 1.17e12T^{2} \)
59 \( 1 + 1.16e5iT - 2.48e12T^{2} \)
61 \( 1 - 8.51e5T + 3.14e12T^{2} \)
67 \( 1 + 2.23e6iT - 6.06e12T^{2} \)
71 \( 1 - 2.24e6iT - 9.09e12T^{2} \)
73 \( 1 - 6.16e6iT - 1.10e13T^{2} \)
79 \( 1 + 2.92e6T + 1.92e13T^{2} \)
83 \( 1 + 5.41e6iT - 2.71e13T^{2} \)
89 \( 1 + 8.84e6iT - 4.42e13T^{2} \)
97 \( 1 + 8.49e6iT - 8.07e13T^{2} \)
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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

−12.82025575949234795311447988965, −11.52685177640953107801689553158, −10.95305258679033364806876662735, −9.943916189151955181786300780353, −8.848763945724985909014369263205, −7.44676025834169670867715401818, −5.99443203019118863746951757221, −3.66158168816572683693105705613, −2.92692397971545992475446099532, −2.07150785074754413236252921880, 0.13760953088046390205071616138, 2.19088009640744701885683644075, 4.53282843046565857137052786374, 5.18103278993477356446975979700, 6.93580873693844367636729529888, 7.83255396716885169471364206528, 8.917481841227256820412307487248, 9.420128316023307646828885420183, 11.51368133715486905287695951677, 12.93918027751364777643151533909

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