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Properties

Label 2-91-13.12-c7-0-25
Degree $2$
Conductor $91$
Sign $-0.611 - 0.790i$
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  + 10.6i·2-s + 74.1·3-s + 14.8·4-s + 114. i·5-s + 788. i·6-s + 343i·7-s + 1.51e3i·8-s + 3.30e3·9-s − 1.21e3·10-s + 168. i·11-s + 1.10e3·12-s + (−6.26e3 + 4.84e3i)13-s − 3.64e3·14-s + 8.46e3i·15-s − 1.42e4·16-s − 1.38e4·17-s + ⋯
L(s)  = 1  + 0.940i·2-s + 1.58·3-s + 0.116·4-s + 0.408i·5-s + 1.48i·6-s + 0.377i·7-s + 1.04i·8-s + 1.51·9-s − 0.384·10-s + 0.0381i·11-s + 0.184·12-s + (−0.790 + 0.611i)13-s − 0.355·14-s + 0.647i·15-s − 0.870·16-s − 0.684·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.790i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.611 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.611 - 0.790i$
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: \((2,\ 91,\ (\ :7/2),\ -0.611 - 0.790i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.64117 + 3.34511i\)
\(L(\frac12)\) \(\approx\) \(1.64117 + 3.34511i\)
\(L(\frac{9}{2})\) 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 \( 1 + (6.26e3 - 4.84e3i)T \)
good2 \( 1 - 10.6iT - 128T^{2} \)
3 \( 1 - 74.1T + 2.18e3T^{2} \)
5 \( 1 - 114. iT - 7.81e4T^{2} \)
11 \( 1 - 168. iT - 1.94e7T^{2} \)
17 \( 1 + 1.38e4T + 4.10e8T^{2} \)
19 \( 1 + 939. iT - 8.93e8T^{2} \)
23 \( 1 - 5.33e4T + 3.40e9T^{2} \)
29 \( 1 - 1.43e5T + 1.72e10T^{2} \)
31 \( 1 - 6.85e4iT - 2.75e10T^{2} \)
37 \( 1 - 6.83e4iT - 9.49e10T^{2} \)
41 \( 1 + 3.35e5iT - 1.94e11T^{2} \)
43 \( 1 - 2.49e5T + 2.71e11T^{2} \)
47 \( 1 + 6.14e5iT - 5.06e11T^{2} \)
53 \( 1 + 1.14e6T + 1.17e12T^{2} \)
59 \( 1 - 2.21e6iT - 2.48e12T^{2} \)
61 \( 1 - 5.94e5T + 3.14e12T^{2} \)
67 \( 1 + 2.83e6iT - 6.06e12T^{2} \)
71 \( 1 + 4.83e6iT - 9.09e12T^{2} \)
73 \( 1 - 2.87e5iT - 1.10e13T^{2} \)
79 \( 1 - 7.55e6T + 1.92e13T^{2} \)
83 \( 1 + 1.78e6iT - 2.71e13T^{2} \)
89 \( 1 - 4.47e4iT - 4.42e13T^{2} \)
97 \( 1 - 2.21e6iT - 8.07e13T^{2} \)
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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

−13.57276469216163675367577718670, −12.16637820489924862214452027532, −10.73652049521355524020896020952, −9.257953586613195957928114299728, −8.505518444055057641142316273573, −7.39931018226523155437331091533, −6.60058987280959642908721388608, −4.80223777624443454967613227962, −3.00349946339595740431663999882, −2.08780230052101397137509552187, 0.974320155817078124027491527328, 2.36856645373127242437549687566, 3.21433137535400644708194234030, 4.54253285855022821297423390154, 6.87623916832377815111597002910, 8.004829505705143324301022745528, 9.148725511639480159087484792560, 10.01437573073218807736324052387, 11.12128006364818217406363871799, 12.57883630378249615830423972609

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