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Properties

Label 2-91-13.3-c1-0-1
Degree $2$
Conductor $91$
Sign $-0.183 - 0.983i$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.37 + 2.37i)2-s + (−0.682 − 1.18i)3-s + (−2.75 + 4.77i)4-s + 0.741·5-s + (1.87 − 3.23i)6-s + (0.5 − 0.866i)7-s − 9.63·8-s + (0.568 − 0.984i)9-s + (1.01 + 1.75i)10-s + (0.682 + 1.18i)11-s + 7.52·12-s + (0.301 − 3.59i)13-s + 2.74·14-s + (−0.505 − 0.875i)15-s + (−7.68 − 13.3i)16-s + (2.07 − 3.59i)17-s + ⋯
L(s)  = 1  + (0.969 + 1.67i)2-s + (−0.393 − 0.682i)3-s + (−1.37 + 2.38i)4-s + 0.331·5-s + (0.763 − 1.32i)6-s + (0.188 − 0.327i)7-s − 3.40·8-s + (0.189 − 0.328i)9-s + (0.321 + 0.556i)10-s + (0.205 + 0.356i)11-s + 2.17·12-s + (0.0837 − 0.996i)13-s + 0.732·14-s + (−0.130 − 0.226i)15-s + (−1.92 − 3.32i)16-s + (0.503 − 0.871i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.183 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.183 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.183 - 0.983i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.183 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.843394 + 1.01522i\)
\(L(\frac12)\) \(\approx\) \(0.843394 + 1.01522i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.301 + 3.59i)T \)
good2 \( 1 + (-1.37 - 2.37i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.682 + 1.18i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.741T + 5T^{2} \)
11 \( 1 + (-0.682 - 1.18i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.07 + 3.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.63 - 6.29i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.16 - 2.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.203 - 0.353i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.77T + 31T^{2} \)
37 \( 1 + (-3.05 - 5.28i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.627 + 1.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.870 + 1.50i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.85T + 47T^{2} \)
53 \( 1 - 4.56T + 53T^{2} \)
59 \( 1 + (-5.49 + 9.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.26 - 5.65i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.87 - 11.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.40 + 4.17i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.06T + 73T^{2} \)
79 \( 1 + 9.12T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + (-0.880 - 1.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.76 - 8.25i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.47370370590669767144398984791, −13.44196758512014609865290821627, −12.70740277457913495694297792110, −11.84971749814349828662481671324, −9.742634689799937751570946455503, −8.153719180867900801146071408757, −7.30128235171785117719689606894, −6.26577511137234972210479521632, −5.34043556153575592319343271103, −3.76989902692039234658638743764, 2.10921035800543073282570783096, 3.94484141228716150011905692849, 4.93616531731032894769853046186, 6.11572720273219881002523338681, 8.932890784200642406622662569011, 9.887294130874420909763038492984, 10.90993065617732156947813774452, 11.45472148396085935432940401502, 12.67285821759755872885701101758, 13.53327415326027890108773734179

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