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Properties

Label 2-91-91.81-c1-0-4
Degree $2$
Conductor $91$
Sign $-0.803 + 0.595i$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s − 3·3-s + (0.500 + 0.866i)4-s + (−1.5 − 2.59i)5-s + (1.5 − 2.59i)6-s + (−2.5 − 0.866i)7-s − 3·8-s + 6·9-s + 3·10-s − 3·11-s + (−1.50 − 2.59i)12-s + (−1 + 3.46i)13-s + (2 − 1.73i)14-s + (4.5 + 7.79i)15-s + (0.500 − 0.866i)16-s + (1 + 1.73i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s − 1.73·3-s + (0.250 + 0.433i)4-s + (−0.670 − 1.16i)5-s + (0.612 − 1.06i)6-s + (−0.944 − 0.327i)7-s − 1.06·8-s + 2·9-s + 0.948·10-s − 0.904·11-s + (−0.433 − 0.749i)12-s + (−0.277 + 0.960i)13-s + (0.534 − 0.462i)14-s + (1.16 + 2.01i)15-s + (0.125 − 0.216i)16-s + (0.242 + 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.803 + 0.595i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.803 + 0.595i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.5 + 0.866i)T \)
13 \( 1 + (1 - 3.46i)T \)
good2 \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + 3T + 3T^{2} \)
5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.5 + 6.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 + 3T + 67T^{2} \)
71 \( 1 + (6.5 - 11.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.5 + 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.17083072304890003935458042810, −12.35600926309536103741998797111, −11.82021533416337375398473062326, −10.54540591994848998103598344623, −9.180931829567947556896723749772, −7.74588984405759713913977021044, −6.70721235102570303894165474320, −5.59003972084021475716683567771, −4.19143906619076892162692347148, 0, 3.04189453050271718744975784464, 5.38223469062928345016302182981, 6.33600266034391586792357201625, 7.37774941531169181062342851270, 9.710119850407055439006879556857, 10.61928492568430797996677418294, 11.03561694622216597655332642963, 12.08027994337324273952628204747, 12.87148941030843382217326914293

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