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Properties

Label 2-91-13.12-c7-0-12
Degree $2$
Conductor $91$
Sign $-0.326 - 0.945i$
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  + 4.26i·2-s − 55.3·3-s + 109.·4-s + 130. i·5-s − 235. i·6-s − 343i·7-s + 1.01e3i·8-s + 873.·9-s − 556.·10-s + 588. i·11-s − 6.07e3·12-s + (7.48e3 − 2.58e3i)13-s + 1.46e3·14-s − 7.22e3i·15-s + 9.74e3·16-s + 1.04e4·17-s + ⋯
L(s)  = 1  + 0.376i·2-s − 1.18·3-s + 0.858·4-s + 0.467i·5-s − 0.445i·6-s − 0.377i·7-s + 0.699i·8-s + 0.399·9-s − 0.176·10-s + 0.133i·11-s − 1.01·12-s + (0.945 − 0.326i)13-s + 0.142·14-s − 0.553i·15-s + 0.594·16-s + 0.517·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.326 - 0.945i$
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: \((2,\ 91,\ (\ :7/2),\ -0.326 - 0.945i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.779967 + 1.09505i\)
\(L(\frac12)\) \(\approx\) \(0.779967 + 1.09505i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-7.48e3 + 2.58e3i)T \)
good2 \( 1 - 4.26iT - 128T^{2} \)
3 \( 1 + 55.3T + 2.18e3T^{2} \)
5 \( 1 - 130. iT - 7.81e4T^{2} \)
11 \( 1 - 588. iT - 1.94e7T^{2} \)
17 \( 1 - 1.04e4T + 4.10e8T^{2} \)
19 \( 1 + 1.70e4iT - 8.93e8T^{2} \)
23 \( 1 + 4.50e4T + 3.40e9T^{2} \)
29 \( 1 + 2.15e5T + 1.72e10T^{2} \)
31 \( 1 - 2.83e5iT - 2.75e10T^{2} \)
37 \( 1 - 5.86e5iT - 9.49e10T^{2} \)
41 \( 1 - 7.38e5iT - 1.94e11T^{2} \)
43 \( 1 - 2.07e4T + 2.71e11T^{2} \)
47 \( 1 + 4.59e5iT - 5.06e11T^{2} \)
53 \( 1 + 1.50e6T + 1.17e12T^{2} \)
59 \( 1 - 4.27e4iT - 2.48e12T^{2} \)
61 \( 1 - 1.66e6T + 3.14e12T^{2} \)
67 \( 1 - 1.46e6iT - 6.06e12T^{2} \)
71 \( 1 + 2.28e6iT - 9.09e12T^{2} \)
73 \( 1 - 5.71e6iT - 1.10e13T^{2} \)
79 \( 1 - 3.70e6T + 1.92e13T^{2} \)
83 \( 1 - 6.44e6iT - 2.71e13T^{2} \)
89 \( 1 + 1.16e6iT - 4.42e13T^{2} \)
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

−12.80446164690604963336298836724, −11.61436677633440258502542727136, −11.02922399939331654824246751586, −10.16819878500894254584858534821, −8.247069668134980881697502622241, −6.94527899070112417361049720598, −6.22122927561801963114859869029, −5.12215946840341829131153565864, −3.17106394894731475640436977334, −1.26438659950229625376141860298, 0.53364246266703379443372606340, 1.91744620986556153763745518665, 3.78034523051058034161478914245, 5.57610492437850702675670268455, 6.21113157225039558509682536326, 7.67543681472103119553744349499, 9.228166005145891980443822003409, 10.62324522024309655404620360652, 11.31231123956445925604884004241, 12.13795517493712575107813025741

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