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Properties

Label 2-91-13.9-c1-0-5
Degree $2$
Conductor $91$
Sign $0.822 + 0.568i$
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  + (−0.115 + 0.200i)2-s + (1.66 − 2.87i)3-s + (0.973 + 1.68i)4-s − 2.23·5-s + (0.384 + 0.665i)6-s + (0.5 + 0.866i)7-s − 0.913·8-s + (−4.01 − 6.96i)9-s + (0.258 − 0.447i)10-s + (−1.66 + 2.87i)11-s + 6.46·12-s + (3.40 + 1.19i)13-s − 0.231·14-s + (−3.70 + 6.41i)15-s + (−1.84 + 3.18i)16-s + (0.687 + 1.19i)17-s + ⋯
L(s)  = 1  + (−0.0817 + 0.141i)2-s + (0.959 − 1.66i)3-s + (0.486 + 0.842i)4-s − 0.997·5-s + (0.156 + 0.271i)6-s + (0.188 + 0.327i)7-s − 0.322·8-s + (−1.33 − 2.32i)9-s + (0.0816 − 0.141i)10-s + (−0.500 + 0.867i)11-s + 1.86·12-s + (0.943 + 0.330i)13-s − 0.0618·14-s + (−0.957 + 1.65i)15-s + (−0.460 + 0.797i)16-s + (0.166 + 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.822 + 0.568i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.822 + 0.568i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09941 - 0.343098i\)
\(L(\frac12)\) \(\approx\) \(1.09941 - 0.343098i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-3.40 - 1.19i)T \)
good2 \( 1 + (0.115 - 0.200i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.66 + 2.87i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 2.23T + 5T^{2} \)
11 \( 1 + (1.66 - 2.87i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.687 - 1.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.61 + 2.80i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.419 - 0.726i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.303 + 0.525i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.71T + 31T^{2} \)
37 \( 1 + (0.776 - 1.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.58 + 7.94i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.615 + 1.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.62T + 47T^{2} \)
53 \( 1 - 8.39T + 53T^{2} \)
59 \( 1 + (4.41 + 7.64i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.73 + 4.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.09 + 8.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.60 - 4.51i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 3.96T + 73T^{2} \)
79 \( 1 - 6.45T + 79T^{2} \)
83 \( 1 + 4.64T + 83T^{2} \)
89 \( 1 + (4.56 - 7.90i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.67 - 13.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.78207974729377296298621016257, −12.78351063907466474921411146983, −12.17182714957777213462043583243, −11.29642291855801554575648955137, −8.915719136285082260029157961502, −8.095004957867821633147710496278, −7.43293287273454841364254687502, −6.46496255287246761462898355204, −3.67193402793460650513997232666, −2.23590199528254669182359196927, 3.06429129986526176513509224090, 4.25978974538480053534653495049, 5.68597288456527941638358916005, 7.86676651477109985167543999693, 8.741081848373144348222561625066, 10.03768831527980089678828627421, 10.78261554121269469031982695617, 11.48345058769687307890730478741, 13.58422793346959703032198834412, 14.51247349698249085364381465123

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