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Properties

Label 2-91-13.12-c7-0-32
Degree $2$
Conductor $91$
Sign $-0.365 - 0.930i$
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  + 16.9i·2-s + 92.8·3-s − 160.·4-s − 107. i·5-s + 1.57e3i·6-s − 343i·7-s − 554. i·8-s + 6.43e3·9-s + 1.82e3·10-s + 2.26e3i·11-s − 1.49e4·12-s + (7.37e3 − 2.89e3i)13-s + 5.82e3·14-s − 9.99e3i·15-s − 1.11e4·16-s + 5.15e3·17-s + ⋯
L(s)  = 1  + 1.50i·2-s + 1.98·3-s − 1.25·4-s − 0.384i·5-s + 2.98i·6-s − 0.377i·7-s − 0.382i·8-s + 2.94·9-s + 0.577·10-s + 0.513i·11-s − 2.49·12-s + (0.930 − 0.365i)13-s + 0.567·14-s − 0.764i·15-s − 0.680·16-s + 0.254·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 - 0.930i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.365 - 0.930i$
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: \((2,\ 91,\ (\ :7/2),\ -0.365 - 0.930i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.35213 + 3.45215i\)
\(L(\frac12)\) \(\approx\) \(2.35213 + 3.45215i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-7.37e3 + 2.89e3i)T \)
good2 \( 1 - 16.9iT - 128T^{2} \)
3 \( 1 - 92.8T + 2.18e3T^{2} \)
5 \( 1 + 107. iT - 7.81e4T^{2} \)
11 \( 1 - 2.26e3iT - 1.94e7T^{2} \)
17 \( 1 - 5.15e3T + 4.10e8T^{2} \)
19 \( 1 - 4.07e4iT - 8.93e8T^{2} \)
23 \( 1 + 4.03e4T + 3.40e9T^{2} \)
29 \( 1 - 3.51e4T + 1.72e10T^{2} \)
31 \( 1 - 7.70e4iT - 2.75e10T^{2} \)
37 \( 1 + 2.08e5iT - 9.49e10T^{2} \)
41 \( 1 - 6.41e5iT - 1.94e11T^{2} \)
43 \( 1 + 4.51e5T + 2.71e11T^{2} \)
47 \( 1 + 3.36e5iT - 5.06e11T^{2} \)
53 \( 1 + 7.61e5T + 1.17e12T^{2} \)
59 \( 1 + 2.41e6iT - 2.48e12T^{2} \)
61 \( 1 + 1.79e6T + 3.14e12T^{2} \)
67 \( 1 + 3.90e5iT - 6.06e12T^{2} \)
71 \( 1 + 4.94e6iT - 9.09e12T^{2} \)
73 \( 1 + 1.75e6iT - 1.10e13T^{2} \)
79 \( 1 + 2.39e6T + 1.92e13T^{2} \)
83 \( 1 + 4.47e6iT - 2.71e13T^{2} \)
89 \( 1 + 1.02e7iT - 4.42e13T^{2} \)
97 \( 1 - 1.75e7iT - 8.07e13T^{2} \)
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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

−13.54036866513087264413856150695, −12.61535428804102562427393580259, −10.24123600783556073887343168528, −9.151000609205141498599178628316, −8.208789071518912998514239300975, −7.71597712908445489097875884615, −6.47661400572345016350948415761, −4.66313322100528107201290318061, −3.44247971272409825159311937742, −1.62321781613201777945795263887, 1.26323613939469789214910457671, 2.47918396816669830041494305040, 3.21223761404833562779189255332, 4.26386947141631517824418471973, 6.91172037417796925602804269726, 8.443609044644928109089560232302, 9.106203315916038376950909824079, 10.09695239957227044100672371700, 11.15068510900927737184027311044, 12.48285104212077209435185405603

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