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Properties

Label 2-91-13.12-c7-0-43
Degree $2$
Conductor $91$
Sign $-0.692 + 0.721i$
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  + 1.59i·2-s + 7.38·3-s + 125.·4-s − 139. i·5-s + 11.7i·6-s − 343i·7-s + 404. i·8-s − 2.13e3·9-s + 223.·10-s − 5.01e3i·11-s + 926.·12-s + (−5.71e3 − 5.48e3i)13-s + 547.·14-s − 1.03e3i·15-s + 1.54e4·16-s − 3.08e4·17-s + ⋯
L(s)  = 1  + 0.141i·2-s + 0.157·3-s + 0.980·4-s − 0.500i·5-s + 0.0222i·6-s − 0.377i·7-s + 0.279i·8-s − 0.975·9-s + 0.0706·10-s − 1.13i·11-s + 0.154·12-s + (−0.721 − 0.692i)13-s + 0.0533·14-s − 0.0790i·15-s + 0.940·16-s − 1.52·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.692 + 0.721i$
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: \((2,\ 91,\ (\ :7/2),\ -0.692 + 0.721i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.484423 - 1.13638i\)
\(L(\frac12)\) \(\approx\) \(0.484423 - 1.13638i\)
\(L(\frac{9}{2})\) 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 \( 1 + (5.71e3 + 5.48e3i)T \)
good2 \( 1 - 1.59iT - 128T^{2} \)
3 \( 1 - 7.38T + 2.18e3T^{2} \)
5 \( 1 + 139. iT - 7.81e4T^{2} \)
11 \( 1 + 5.01e3iT - 1.94e7T^{2} \)
17 \( 1 + 3.08e4T + 4.10e8T^{2} \)
19 \( 1 - 4.31e4iT - 8.93e8T^{2} \)
23 \( 1 + 4.83e4T + 3.40e9T^{2} \)
29 \( 1 + 1.83e4T + 1.72e10T^{2} \)
31 \( 1 + 1.07e5iT - 2.75e10T^{2} \)
37 \( 1 + 2.43e5iT - 9.49e10T^{2} \)
41 \( 1 + 3.84e4iT - 1.94e11T^{2} \)
43 \( 1 + 9.29e5T + 2.71e11T^{2} \)
47 \( 1 + 1.31e6iT - 5.06e11T^{2} \)
53 \( 1 - 3.19e4T + 1.17e12T^{2} \)
59 \( 1 - 1.28e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.51e6T + 3.14e12T^{2} \)
67 \( 1 + 4.11e6iT - 6.06e12T^{2} \)
71 \( 1 + 4.77e6iT - 9.09e12T^{2} \)
73 \( 1 - 2.13e6iT - 1.10e13T^{2} \)
79 \( 1 + 2.06e4T + 1.92e13T^{2} \)
83 \( 1 - 4.00e6iT - 2.71e13T^{2} \)
89 \( 1 - 2.08e6iT - 4.42e13T^{2} \)
97 \( 1 - 3.67e6iT - 8.07e13T^{2} \)
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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

−12.14867741584925478162552277758, −11.23631989680314328365844982497, −10.27747829210984086012047323418, −8.673004139090273378116116011805, −7.85795983609460593249095855008, −6.41307792909564459492883782415, −5.39109811455038664215262078673, −3.49218911411923323415065124196, −2.13158209769744410073742651664, −0.33160717071462222726549958520, 2.06081713906154816020087564635, 2.84032417289833679331454288212, 4.76519662156146593203191797322, 6.46467491262243603810645564879, 7.15802465878206304476293637234, 8.682277765582693216348047537340, 9.896066480182033961600566130655, 11.16255835051954334424950176538, 11.74819810666577890440200459439, 12.92644033894393020668435804880

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