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Properties

Label 2-91-91.73-c1-0-3
Degree $2$
Conductor $91$
Sign $0.586 - 0.809i$
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  + (−0.186 + 0.694i)2-s + (1.44 + 0.832i)3-s + (1.28 + 0.741i)4-s + (−1.87 − 0.501i)5-s + (−0.846 + 0.846i)6-s + (−0.783 − 2.52i)7-s + (−1.77 + 1.77i)8-s + (−0.115 − 0.199i)9-s + (0.696 − 1.20i)10-s + (0.825 + 3.08i)11-s + (1.23 + 2.13i)12-s + (0.846 − 3.50i)13-s + (1.90 − 0.0737i)14-s + (−2.27 − 2.27i)15-s + (0.582 + 1.00i)16-s + (−0.254 + 0.440i)17-s + ⋯
L(s)  = 1  + (−0.131 + 0.491i)2-s + (0.832 + 0.480i)3-s + (0.642 + 0.370i)4-s + (−0.836 − 0.224i)5-s + (−0.345 + 0.345i)6-s + (−0.296 − 0.955i)7-s + (−0.626 + 0.626i)8-s + (−0.0383 − 0.0664i)9-s + (0.220 − 0.381i)10-s + (0.248 + 0.929i)11-s + (0.356 + 0.617i)12-s + (0.234 − 0.972i)13-s + (0.508 − 0.0196i)14-s + (−0.588 − 0.588i)15-s + (0.145 + 0.252i)16-s + (−0.0616 + 0.106i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.586 - 0.809i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.586 - 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00870 + 0.514690i\)
\(L(\frac12)\) \(\approx\) \(1.00870 + 0.514690i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.783 + 2.52i)T \)
13 \( 1 + (-0.846 + 3.50i)T \)
good2 \( 1 + (0.186 - 0.694i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (-1.44 - 0.832i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.87 + 0.501i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.825 - 3.08i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.254 - 0.440i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.65 + 0.710i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.49 + 1.44i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.40T + 29T^{2} \)
31 \( 1 + (-0.827 - 3.08i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-9.40 - 2.51i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (5.34 - 5.34i)T - 41iT^{2} \)
43 \( 1 - 12.5iT - 43T^{2} \)
47 \( 1 + (-2.88 + 10.7i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.42 - 5.93i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.74 + 1.00i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.51 - 3.18i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.56 + 1.75i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.90 + 1.90i)T + 71iT^{2} \)
73 \( 1 + (-0.252 + 0.0676i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.78 + 4.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.86 + 5.86i)T - 83iT^{2} \)
89 \( 1 + (3.17 - 11.8i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (7.04 - 7.04i)T - 97iT^{2} \)
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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

−14.80292080844686205022738967904, −13.22918284032447814842878245712, −12.17717769375022466593707813142, −10.98645210714581324505699377288, −9.755452707965743303312141420914, −8.403752535407163550707196328952, −7.66538666066451828871162671310, −6.49245335990932520288424109235, −4.30099785514033642316561427168, −3.09913730814108839314085600943, 2.21362794080358332754646848344, 3.50325818085472213004741956819, 5.93064762814937713879289629354, 7.18802244639109262773244850977, 8.472357361334682222722835262354, 9.381597741520285439776506929543, 11.06077652115965614360266621526, 11.61854277382875134182953773036, 12.74522549697739580685217252341, 13.98643847616914565881905080353

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