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Properties

Label 2-91-13.3-c1-0-3
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: $\chi_{91} (29, \cdot )$
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

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

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