Properties

Label 2-91-13.12-c7-0-31
Degree $2$
Conductor $91$
Sign $0.867 - 0.497i$
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  − 3.42i·2-s + 69.6·3-s + 116.·4-s + 307. i·5-s − 238. i·6-s + 343i·7-s − 837. i·8-s + 2.66e3·9-s + 1.05e3·10-s + 4.37e3i·11-s + 8.09e3·12-s + (−3.94e3 − 6.86e3i)13-s + 1.17e3·14-s + 2.14e4i·15-s + 1.20e4·16-s + 3.77e4·17-s + ⋯
L(s)  = 1  − 0.303i·2-s + 1.48·3-s + 0.908·4-s + 1.10i·5-s − 0.451i·6-s + 0.377i·7-s − 0.578i·8-s + 1.21·9-s + 0.333·10-s + 0.991i·11-s + 1.35·12-s + (−0.497 − 0.867i)13-s + 0.114·14-s + 1.64i·15-s + 0.732·16-s + 1.86·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.867 - 0.497i$
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: \((2,\ 91,\ (\ :7/2),\ 0.867 - 0.497i)\)

Particular Values

\(L(4)\) \(\approx\) \(4.04157 + 1.07775i\)
\(L(\frac12)\) \(\approx\) \(4.04157 + 1.07775i\)
\(L(\frac{9}{2})\) 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 \( 1 + (3.94e3 + 6.86e3i)T \)
good2 \( 1 + 3.42iT - 128T^{2} \)
3 \( 1 - 69.6T + 2.18e3T^{2} \)
5 \( 1 - 307. iT - 7.81e4T^{2} \)
11 \( 1 - 4.37e3iT - 1.94e7T^{2} \)
17 \( 1 - 3.77e4T + 4.10e8T^{2} \)
19 \( 1 - 3.01e4iT - 8.93e8T^{2} \)
23 \( 1 + 1.04e5T + 3.40e9T^{2} \)
29 \( 1 - 8.25e3T + 1.72e10T^{2} \)
31 \( 1 + 1.51e5iT - 2.75e10T^{2} \)
37 \( 1 - 1.28e5iT - 9.49e10T^{2} \)
41 \( 1 - 6.25e5iT - 1.94e11T^{2} \)
43 \( 1 - 4.43e5T + 2.71e11T^{2} \)
47 \( 1 + 7.09e5iT - 5.06e11T^{2} \)
53 \( 1 - 7.94e5T + 1.17e12T^{2} \)
59 \( 1 + 1.32e5iT - 2.48e12T^{2} \)
61 \( 1 - 1.09e6T + 3.14e12T^{2} \)
67 \( 1 + 3.28e6iT - 6.06e12T^{2} \)
71 \( 1 + 7.75e5iT - 9.09e12T^{2} \)
73 \( 1 + 5.94e6iT - 1.10e13T^{2} \)
79 \( 1 + 6.90e6T + 1.92e13T^{2} \)
83 \( 1 + 2.00e6iT - 2.71e13T^{2} \)
89 \( 1 - 1.06e6iT - 4.42e13T^{2} \)
97 \( 1 + 1.07e7iT - 8.07e13T^{2} \)
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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

−12.64998844609153121759679435868, −11.83148194266452275021139944928, −10.10806229905807716670889381223, −9.959718254925337884969566008533, −7.931921973174919014317034430065, −7.50631196337417211678973072587, −6.00009888472450057745169286004, −3.61034874548469531715982565237, −2.76769888292958077283859109760, −1.84576191010529481200185279027, 1.18862760649827074587287255980, 2.55184334813339896498383615039, 3.87188846900590879954227870521, 5.58143497283447281223137624670, 7.22989831194697689241697910638, 8.140443446319824917430136565941, 8.969236717614221629937900501093, 10.15048242955914241800459247269, 11.69916140992651131359051412923, 12.68765407668210038841443147684

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