Properties

Label 2-91-7.4-c1-0-4
Degree $2$
Conductor $91$
Sign $0.266 + 0.963i$
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  + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s + (0.5 − 2.59i)7-s − 3·8-s + (1.5 + 2.59i)9-s + (1.5 − 2.59i)11-s − 13-s + (−2.5 + 0.866i)14-s + (0.500 + 0.866i)16-s + (−3.5 + 6.06i)17-s + (1.5 − 2.59i)18-s + (3.5 + 6.06i)19-s − 3·22-s + (3 + 5.19i)23-s + (2.5 − 4.33i)25-s + (0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s + (0.188 − 0.981i)7-s − 1.06·8-s + (0.5 + 0.866i)9-s + (0.452 − 0.783i)11-s − 0.277·13-s + (−0.668 + 0.231i)14-s + (0.125 + 0.216i)16-s + (−0.848 + 1.47i)17-s + (0.353 − 0.612i)18-s + (0.802 + 1.39i)19-s − 0.639·22-s + (0.625 + 1.08i)23-s + (0.5 − 0.866i)25-s + (0.0980 + 0.169i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.266 + 0.963i$
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: \((2,\ 91,\ (\ :1/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.710985 - 0.540885i\)
\(L(\frac12)\) \(\approx\) \(0.710985 - 0.540885i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.5 + 2.59i)T \)
13 \( 1 + T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.5 - 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (3.5 + 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5T + 71T^{2} \)
73 \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−13.87282216963460635583314164429, −12.75164762652578582436312757752, −11.37304074217237000786930589773, −10.65772235316585565648228861879, −9.849000502889582452239528035307, −8.406693231176218749553958013660, −7.06337995585277557712301026186, −5.62600571695683110288567633757, −3.80222178285826781011550157876, −1.63605626214557679743548316152, 2.83477923611271154726898378845, 4.89464592389451657134899984228, 6.63224762833152679414714385275, 7.30209499879233676382938122844, 8.991797058531261839995299601227, 9.353677272744523833793081283445, 11.39858744034346832621337418871, 12.10984816238046023892656484342, 13.10547305650341557205541427777, 14.80902644681530851113949964493

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