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Properties

Label 2-91-91.30-c1-0-4
Degree $2$
Conductor $91$
Sign $-0.794 + 0.606i$
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  + (−1.5 + 0.866i)2-s − 3-s + (0.5 − 0.866i)4-s + (−1.5 − 0.866i)5-s + (1.5 − 0.866i)6-s + (−2.5 + 0.866i)7-s − 1.73i·8-s − 2·9-s + 3·10-s − 5.19i·11-s + (−0.5 + 0.866i)12-s + (−1 + 3.46i)13-s + (3 − 3.46i)14-s + (1.5 + 0.866i)15-s + (2.49 + 4.33i)16-s + (−3 + 5.19i)17-s + ⋯
L(s)  = 1  + (−1.06 + 0.612i)2-s − 0.577·3-s + (0.250 − 0.433i)4-s + (−0.670 − 0.387i)5-s + (0.612 − 0.353i)6-s + (−0.944 + 0.327i)7-s − 0.612i·8-s − 0.666·9-s + 0.948·10-s − 1.56i·11-s + (−0.144 + 0.250i)12-s + (−0.277 + 0.960i)13-s + (0.801 − 0.925i)14-s + (0.387 + 0.223i)15-s + (0.624 + 1.08i)16-s + (−0.727 + 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 + 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.794 + 0.606i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.794 + 0.606i)\)

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 \( 1 + (1.5 - 0.866i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + T + 3T^{2} \)
5 \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 5.19iT - 11T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.5 + 4.33i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 + 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 - 8.66iT - 67T^{2} \)
71 \( 1 + (-1.5 + 0.866i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.5 - 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + (6 - 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.5 - 2.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.63904160802417821624663628691, −12.42643715581397599566395255420, −11.45890042196068561769156560531, −10.25644898687776839236866277489, −8.791491402927148758584956909343, −8.434338348736293880930741680040, −6.76391275547223292903500878979, −5.86769882254247262044039795784, −3.72903192561893816413158387599, 0, 2.84908049331121638651684860775, 4.99121960582268942783276276215, 6.78328450407209927392109956700, 7.88654935467074101815542443262, 9.395539239652842239684726576896, 10.12907394878540675614875167831, 11.21711552819622516178340218458, 11.93305911788362475343826251061, 13.16371348442870399093902062507

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