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Properties

Label 2-91-13.12-c7-0-19
Degree $2$
Conductor $91$
Sign $0.739 + 0.673i$
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  − 21.0i·2-s + 55.1·3-s − 313.·4-s + 298. i·5-s − 1.15e3i·6-s − 343i·7-s + 3.88e3i·8-s + 850.·9-s + 6.26e3·10-s + 3.27e3i·11-s − 1.72e4·12-s + (−5.33e3 + 5.85e3i)13-s − 7.20e3·14-s + 1.64e4i·15-s + 4.15e4·16-s + 3.39e4·17-s + ⋯
L(s)  = 1  − 1.85i·2-s + 1.17·3-s − 2.44·4-s + 1.06i·5-s − 2.18i·6-s − 0.377i·7-s + 2.68i·8-s + 0.388·9-s + 1.98·10-s + 0.742i·11-s − 2.88·12-s + (−0.673 + 0.739i)13-s − 0.701·14-s + 1.25i·15-s + 2.53·16-s + 1.67·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.739 + 0.673i$
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: \((2,\ 91,\ (\ :7/2),\ 0.739 + 0.673i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.07431 - 0.802918i\)
\(L(\frac12)\) \(\approx\) \(2.07431 - 0.802918i\)
\(L(\frac{9}{2})\) 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 \( 1 + (5.33e3 - 5.85e3i)T \)
good2 \( 1 + 21.0iT - 128T^{2} \)
3 \( 1 - 55.1T + 2.18e3T^{2} \)
5 \( 1 - 298. iT - 7.81e4T^{2} \)
11 \( 1 - 3.27e3iT - 1.94e7T^{2} \)
17 \( 1 - 3.39e4T + 4.10e8T^{2} \)
19 \( 1 + 961. iT - 8.93e8T^{2} \)
23 \( 1 - 7.04e4T + 3.40e9T^{2} \)
29 \( 1 - 1.87e5T + 1.72e10T^{2} \)
31 \( 1 - 1.39e5iT - 2.75e10T^{2} \)
37 \( 1 - 496. iT - 9.49e10T^{2} \)
41 \( 1 - 8.11e5iT - 1.94e11T^{2} \)
43 \( 1 + 6.96e5T + 2.71e11T^{2} \)
47 \( 1 + 2.54e5iT - 5.06e11T^{2} \)
53 \( 1 - 4.21e5T + 1.17e12T^{2} \)
59 \( 1 - 1.45e6iT - 2.48e12T^{2} \)
61 \( 1 + 1.30e6T + 3.14e12T^{2} \)
67 \( 1 + 6.54e5iT - 6.06e12T^{2} \)
71 \( 1 - 2.06e6iT - 9.09e12T^{2} \)
73 \( 1 - 4.83e5iT - 1.10e13T^{2} \)
79 \( 1 + 7.21e6T + 1.92e13T^{2} \)
83 \( 1 - 4.74e6iT - 2.71e13T^{2} \)
89 \( 1 + 9.62e6iT - 4.42e13T^{2} \)
97 \( 1 - 9.06e6iT - 8.07e13T^{2} \)
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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

−12.40455613190115337569160535553, −11.45077206394143723367989510403, −10.23385763585660033883869356835, −9.729378509291315578563045674703, −8.510208641457236879485314280273, −7.18297892888268014765748514725, −4.68879787745684748913308269512, −3.27993305130862293448955095217, −2.74392974142945506824610985168, −1.40779539585001623481446115822, 0.67889408688249813291830963773, 3.26573811822048646888772338638, 4.91654088114483884778780378999, 5.77346498781574408657831801915, 7.44039080526743061999595228158, 8.318203598745970912495210992077, 8.833312995567483168800293086343, 9.846431012021360463299511337348, 12.34906457303431593349859177219, 13.29777438546825263759136893407

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