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Properties

Label 2-91-91.9-c1-0-3
Degree $2$
Conductor $91$
Sign $0.900 + 0.434i$
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  + (0.134 + 0.232i)2-s − 1.14·3-s + (0.964 − 1.66i)4-s + (1.28 − 2.21i)5-s + (−0.153 − 0.265i)6-s + (0.773 + 2.53i)7-s + 1.05·8-s − 1.69·9-s + 0.686·10-s + 3.94·11-s + (−1.10 + 1.90i)12-s + (−3.15 + 1.74i)13-s + (−0.483 + 0.518i)14-s + (−1.46 + 2.53i)15-s + (−1.78 − 3.09i)16-s + (−0.392 + 0.679i)17-s + ⋯
L(s)  = 1  + (0.0947 + 0.164i)2-s − 0.659·3-s + (0.482 − 0.834i)4-s + (0.572 − 0.992i)5-s + (−0.0625 − 0.108i)6-s + (0.292 + 0.956i)7-s + 0.372·8-s − 0.564·9-s + 0.217·10-s + 1.18·11-s + (−0.318 + 0.550i)12-s + (−0.874 + 0.484i)13-s + (−0.129 + 0.138i)14-s + (−0.378 + 0.654i)15-s + (−0.446 − 0.773i)16-s + (−0.0952 + 0.164i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.900 + 0.434i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.900 + 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.975834 - 0.222936i\)
\(L(\frac12)\) \(\approx\) \(0.975834 - 0.222936i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.773 - 2.53i)T \)
13 \( 1 + (3.15 - 1.74i)T \)
good2 \( 1 + (-0.134 - 0.232i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + 1.14T + 3T^{2} \)
5 \( 1 + (-1.28 + 2.21i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 3.94T + 11T^{2} \)
17 \( 1 + (0.392 - 0.679i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 7.49T + 19T^{2} \)
23 \( 1 + (-3.97 - 6.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.17 - 2.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.27 - 2.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.37 + 5.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.21 + 2.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.12 - 1.94i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.658 - 1.14i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.63 + 8.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.48 + 7.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 9.44T + 61T^{2} \)
67 \( 1 + 1.35T + 67T^{2} \)
71 \( 1 + (6.15 + 10.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.384 + 0.665i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.09 - 5.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.07T + 83T^{2} \)
89 \( 1 + (3.83 + 6.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.18 - 2.05i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.29610007424518368562952607293, −12.78241516764902391913890075140, −11.82124610428359995933448855516, −11.01451701721186623279864227321, −9.498922178270736048861091683354, −8.761773961348950682193380237474, −6.71360741457434206127121291665, −5.70213113438885450317625325141, −4.89389388606952577196881471963, −1.83464653610401243271167157870, 2.66162183724471948812018366829, 4.35703602722561885347661362994, 6.37067939331133700901296559288, 7.00575749009347387700981669530, 8.496431533441518691760817241402, 10.30812704168354523948377283050, 10.96517216362457356249000421450, 11.88600218551350098302009828527, 12.98612336989371191517653654250, 14.25665149687092050893091800133

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