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Properties

Label 2-91-13.12-c7-0-7
Degree $2$
Conductor $91$
Sign $-0.925 - 0.379i$
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  + 20.4i·2-s − 67.6·3-s − 290.·4-s − 268. i·5-s − 1.38e3i·6-s + 343i·7-s − 3.31e3i·8-s + 2.39e3·9-s + 5.48e3·10-s − 4.03e3i·11-s + 1.96e4·12-s + (−3.00e3 + 7.32e3i)13-s − 7.01e3·14-s + 1.81e4i·15-s + 3.07e4·16-s + 3.67e3·17-s + ⋯
L(s)  = 1  + 1.80i·2-s − 1.44·3-s − 2.26·4-s − 0.960i·5-s − 2.61i·6-s + 0.377i·7-s − 2.29i·8-s + 1.09·9-s + 1.73·10-s − 0.913i·11-s + 3.28·12-s + (−0.379 + 0.925i)13-s − 0.683·14-s + 1.38i·15-s + 1.87·16-s + 0.181·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.925 - 0.379i$
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: \((2,\ 91,\ (\ :7/2),\ -0.925 - 0.379i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.104103 + 0.527567i\)
\(L(\frac12)\) \(\approx\) \(0.104103 + 0.527567i\)
\(L(\frac{9}{2})\) 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 \( 1 + (3.00e3 - 7.32e3i)T \)
good2 \( 1 - 20.4iT - 128T^{2} \)
3 \( 1 + 67.6T + 2.18e3T^{2} \)
5 \( 1 + 268. iT - 7.81e4T^{2} \)
11 \( 1 + 4.03e3iT - 1.94e7T^{2} \)
17 \( 1 - 3.67e3T + 4.10e8T^{2} \)
19 \( 1 + 4.98e4iT - 8.93e8T^{2} \)
23 \( 1 + 4.63e4T + 3.40e9T^{2} \)
29 \( 1 + 8.31e4T + 1.72e10T^{2} \)
31 \( 1 - 1.88e5iT - 2.75e10T^{2} \)
37 \( 1 + 4.97e5iT - 9.49e10T^{2} \)
41 \( 1 - 6.52e5iT - 1.94e11T^{2} \)
43 \( 1 - 1.50e5T + 2.71e11T^{2} \)
47 \( 1 + 3.60e5iT - 5.06e11T^{2} \)
53 \( 1 + 5.24e5T + 1.17e12T^{2} \)
59 \( 1 - 1.52e6iT - 2.48e12T^{2} \)
61 \( 1 + 2.72e6T + 3.14e12T^{2} \)
67 \( 1 - 1.22e6iT - 6.06e12T^{2} \)
71 \( 1 - 2.61e6iT - 9.09e12T^{2} \)
73 \( 1 - 4.01e6iT - 1.10e13T^{2} \)
79 \( 1 - 7.40e6T + 1.92e13T^{2} \)
83 \( 1 - 1.71e6iT - 2.71e13T^{2} \)
89 \( 1 - 1.29e6iT - 4.42e13T^{2} \)
97 \( 1 - 5.54e6iT - 8.07e13T^{2} \)
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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

−13.27486129384843302120493221081, −12.26228665608965435009074433012, −11.12643831938543263971667344327, −9.363918107973907401899030678147, −8.572724441454489568151718399759, −7.12387154534701012844628627504, −6.12936524950851822729810925862, −5.27767312606047905457948823621, −4.50893223565306897411231819087, −0.70546917009070358312371328407, 0.35542750935567555739103539932, 1.84011944855944150334485041406, 3.41154354752505659408504499467, 4.71600932696035284481307120690, 6.05150439966619348004994081614, 7.68637625434932270672942872227, 9.827329405866422455625683765900, 10.36805319555340773852459702220, 11.05747971249004326043102218563, 12.10288724911852905774605454438

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