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Properties

Label 2-93-31.8-c1-0-1
Degree $2$
Conductor $93$
Sign $0.517 - 0.855i$
Analytic cond. $0.742608$
Root an. cond. $0.861747$
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  + (−0.413 + 1.27i)2-s + (−0.309 − 0.951i)3-s + (0.169 + 0.122i)4-s + 0.827·5-s + 1.33·6-s + (2.78 + 2.02i)7-s + (−2.39 + 1.73i)8-s + (−0.809 + 0.587i)9-s + (−0.342 + 1.05i)10-s + (−1.37 − 0.997i)11-s + (0.0646 − 0.198i)12-s + (−0.989 − 3.04i)13-s + (−3.72 + 2.70i)14-s + (−0.255 − 0.786i)15-s + (−1.09 − 3.36i)16-s + (2.01 − 1.46i)17-s + ⋯
L(s)  = 1  + (−0.292 + 0.899i)2-s + (−0.178 − 0.549i)3-s + (0.0845 + 0.0614i)4-s + 0.369·5-s + 0.546·6-s + (1.05 + 0.765i)7-s + (−0.845 + 0.614i)8-s + (−0.269 + 0.195i)9-s + (−0.108 + 0.332i)10-s + (−0.414 − 0.300i)11-s + (0.0186 − 0.0573i)12-s + (−0.274 − 0.844i)13-s + (−0.996 + 0.724i)14-s + (−0.0659 − 0.203i)15-s + (−0.273 − 0.841i)16-s + (0.488 − 0.354i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.517 - 0.855i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.517 - 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.823100 + 0.464318i\)
\(L(\frac12)\) \(\approx\) \(0.823100 + 0.464318i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (4.65 + 3.05i)T \)
good2 \( 1 + (0.413 - 1.27i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 - 0.827T + 5T^{2} \)
7 \( 1 + (-2.78 - 2.02i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (1.37 + 0.997i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.989 + 3.04i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.01 + 1.46i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.755 + 2.32i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (3.78 - 2.75i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-2.82 + 8.70i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 - 1.85T + 37T^{2} \)
41 \( 1 + (0.399 - 1.22i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (4.00 - 12.3i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-2.79 - 8.61i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.62 + 2.63i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.732 - 2.25i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 - 8.32T + 61T^{2} \)
67 \( 1 - 2.47T + 67T^{2} \)
71 \( 1 + (-11.6 + 8.45i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (5.59 + 4.06i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (14.0 - 10.1i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.67 + 5.14i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-6.30 - 4.57i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-10.5 - 7.65i)T + (29.9 + 92.2i)T^{2} \)
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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

−14.43955683433753971870379027397, −13.27948858568371947723746495399, −11.97242005463574820901642907625, −11.29707288727766806009248646223, −9.596637978413851099012998387515, −8.128105588164779249505220493224, −7.72160583362079938571454114357, −6.10146992004326984524751884231, −5.31970507491967699734190408476, −2.47743816100882892276869117312, 1.85125260776948585746469600207, 3.89276293365728293458237296732, 5.41756312244758029125684010884, 7.05483218258356471953625075650, 8.635290788508018598059496020607, 10.01418563072911794048727478059, 10.50254212313266875116613693169, 11.53971177584030727290361778260, 12.42547626641164424558170927800, 14.01064685864983157156607488087

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