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Properties

Label 2-91-91.74-c1-0-3
Degree $2$
Conductor $91$
Sign $0.788 - 0.615i$
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  + 2-s + (1.5 + 2.59i)3-s − 4-s + (−1.5 − 2.59i)5-s + (1.5 + 2.59i)6-s + (2 − 1.73i)7-s − 3·8-s + (−3 + 5.19i)9-s + (−1.5 − 2.59i)10-s + (1.5 + 2.59i)11-s + (−1.5 − 2.59i)12-s + (−1 − 3.46i)13-s + (2 − 1.73i)14-s + (4.5 − 7.79i)15-s − 16-s − 2·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.866 + 1.49i)3-s − 0.5·4-s + (−0.670 − 1.16i)5-s + (0.612 + 1.06i)6-s + (0.755 − 0.654i)7-s − 1.06·8-s + (−1 + 1.73i)9-s + (−0.474 − 0.821i)10-s + (0.452 + 0.783i)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 − 0.250·16-s − 0.485·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.788 - 0.615i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.788 - 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31154 + 0.451563i\)
\(L(\frac12)\) \(\approx\) \(1.31154 + 0.451563i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2 + 1.73i)T \)
13 \( 1 + (1 + 3.46i)T \)
good2 \( 1 - T + 2T^{2} \)
3 \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
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 \( 1 - 2T + 37T^{2} \)
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 \( 1 + 4T + 59T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
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 \( 1 - 6T + 89T^{2} \)
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

−14.46585895822143833311136832592, −13.31546639609939714767137797220, −12.31949065152968956766145001265, −10.90733260216413571841262765518, −9.621496200049852987512287090668, −8.809710892776191299255148903689, −7.87691833304512702463473280831, −5.01391016386276166050628934385, −4.58136042881269318009410929438, −3.55245046201534053927398399933, 2.46899221090451149875695901858, 3.86124221618274047271406154688, 5.99997245797369236524473297651, 7.13166786646256717953807301980, 8.239790492814534830455196396157, 9.145286917569303267357955186980, 11.42096805350290347665023111908, 11.95064385211965783578853288240, 13.16586848332719005533561895932, 14.11229235269877279002746093897

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