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Properties

Label 2-91-91.76-c1-0-0
Degree $2$
Conductor $91$
Sign $-0.811 + 0.584i$
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  + (−0.607 + 2.26i)2-s + (−2.39 + 1.38i)3-s + (−3.03 − 1.75i)4-s + (1.12 − 1.12i)5-s + (−1.67 − 6.25i)6-s + (−0.437 + 2.60i)7-s + (2.50 − 2.50i)8-s + (2.30 − 4.00i)9-s + (1.87 + 3.23i)10-s + (−3.03 − 0.813i)11-s + 9.68·12-s + (−1.04 + 3.44i)13-s + (−5.64 − 2.57i)14-s + (−1.13 + 4.24i)15-s + (0.649 + 1.12i)16-s + (0.320 − 0.555i)17-s + ⋯
L(s)  = 1  + (−0.429 + 1.60i)2-s + (−1.38 + 0.796i)3-s + (−1.51 − 0.877i)4-s + (0.503 − 0.503i)5-s + (−0.684 − 2.55i)6-s + (−0.165 + 0.986i)7-s + (0.886 − 0.886i)8-s + (0.769 − 1.33i)9-s + (0.591 + 1.02i)10-s + (−0.914 − 0.245i)11-s + 2.79·12-s + (−0.290 + 0.956i)13-s + (−1.51 − 0.689i)14-s + (−0.293 + 1.09i)15-s + (0.162 + 0.281i)16-s + (0.0778 − 0.134i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.811 + 0.584i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.811 + 0.584i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128482 - 0.398018i\)
\(L(\frac12)\) \(\approx\) \(0.128482 - 0.398018i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.437 - 2.60i)T \)
13 \( 1 + (1.04 - 3.44i)T \)
good2 \( 1 + (0.607 - 2.26i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (2.39 - 1.38i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.12 + 1.12i)T - 5iT^{2} \)
11 \( 1 + (3.03 + 0.813i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.320 + 0.555i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.04 - 7.61i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.126 - 0.0730i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.49 - 2.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.73 + 4.73i)T - 31iT^{2} \)
37 \( 1 + (-3.75 - 1.00i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.60 - 1.50i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.42 - 1.40i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.22 + 2.22i)T + 47iT^{2} \)
53 \( 1 + 7.32T + 53T^{2} \)
59 \( 1 + (4.00 - 1.07i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.90 + 2.25i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.366 - 1.36i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-13.8 + 3.70i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.99 - 4.99i)T + 73iT^{2} \)
79 \( 1 + 0.632T + 79T^{2} \)
83 \( 1 + (-1.07 + 1.07i)T - 83iT^{2} \)
89 \( 1 + (-3.51 + 13.1i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.0487 - 0.181i)T + (-84.0 + 48.5i)T^{2} \)
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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

−15.22166557708707324083170226781, −14.08030119155179998884426395881, −12.58395748255565605825914922602, −11.48172574909536359750226737039, −9.959023581913590165132031797090, −9.288940436957970840419645795389, −7.944576684215764238269571105055, −6.27074715885515503981777935475, −5.64207069928988648628146828255, −4.79806788931418410500684654594, 0.69568648588983374142603962967, 2.69832636322815728440675340999, 4.86664375032129609007297281374, 6.49426244525821635242296402929, 7.74191204649014390845982673392, 9.739427806516304449452815376376, 10.64144123206956189201734058192, 11.04555539496183665756004716213, 12.27855546702956916713322092552, 13.01157691192928602539784735140

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