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Properties

Label 2-91-7.4-c1-0-6
Degree $2$
Conductor $91$
Sign $-0.0513 + 0.998i$
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  + (−0.632 − 1.09i)2-s + (1.31 − 2.27i)3-s + (0.199 − 0.344i)4-s + (1.45 + 2.51i)5-s − 3.32·6-s + (−1.29 + 2.30i)7-s − 3.03·8-s + (−1.95 − 3.37i)9-s + (1.83 − 3.18i)10-s + (−1.01 + 1.76i)11-s + (−0.523 − 0.906i)12-s + 13-s + (3.34 − 0.0400i)14-s + 7.62·15-s + (1.52 + 2.63i)16-s + (−1.99 + 3.46i)17-s + ⋯
L(s)  = 1  + (−0.447 − 0.775i)2-s + (0.758 − 1.31i)3-s + (0.0995 − 0.172i)4-s + (0.649 + 1.12i)5-s − 1.35·6-s + (−0.489 + 0.871i)7-s − 1.07·8-s + (−0.650 − 1.12i)9-s + (0.580 − 1.00i)10-s + (−0.307 + 0.531i)11-s + (−0.151 − 0.261i)12-s + 0.277·13-s + (0.894 − 0.0107i)14-s + 1.96·15-s + (0.380 + 0.659i)16-s + (−0.484 + 0.839i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0513 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0513 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.0513 + 0.998i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.0513 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.704926 - 0.742129i\)
\(L(\frac12)\) \(\approx\) \(0.704926 - 0.742129i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.29 - 2.30i)T \)
13 \( 1 - T \)
good2 \( 1 + (0.632 + 1.09i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.31 + 2.27i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.45 - 2.51i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.01 - 1.76i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.99 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.48 + 6.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.313 - 0.543i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.09T + 29T^{2} \)
31 \( 1 + (-5.21 + 9.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.54 - 2.67i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.521T + 41T^{2} \)
43 \( 1 - 0.329T + 43T^{2} \)
47 \( 1 + (5.27 + 9.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.55 - 6.16i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.01 - 1.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.20 + 2.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.34 - 12.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.60T + 71T^{2} \)
73 \( 1 + (1.48 - 2.57i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.38 - 7.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 + (-1.34 - 2.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.32T + 97T^{2} \)
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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

−13.54569818539315467539018983891, −12.82949986531904994234632257471, −11.68032019869949282574962877610, −10.55649675326800644554919584106, −9.478256401013718356347095440274, −8.409231874124545282989890596373, −6.78867062152226493383771502828, −6.16618870871093693955849088181, −2.75840817566795589164627141121, −2.14399081386139421756398149636, 3.31649817649868200497682542822, 4.76004094709702901045631407632, 6.31247818439630753797976588304, 8.019646354919023704901724337398, 8.860963115613241450262454670371, 9.634577012619114911015831440884, 10.67245089792033294449112541744, 12.45606442746698100107136092294, 13.59020095947554589876826521336, 14.52468631515261277184444002665

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