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Properties

Label 2-93-93.89-c1-0-4
Degree $2$
Conductor $93$
Sign $0.994 - 0.106i$
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  + (0.865 − 0.281i)2-s + (1.31 + 1.12i)3-s + (−0.948 + 0.688i)4-s − 2.95i·5-s + (1.45 + 0.602i)6-s + (0.214 − 0.156i)7-s + (−1.69 + 2.33i)8-s + (0.470 + 2.96i)9-s + (−0.831 − 2.56i)10-s + (−0.696 + 0.506i)11-s + (−2.02 − 0.158i)12-s + (−2.76 − 0.897i)13-s + (0.141 − 0.195i)14-s + (3.32 − 3.89i)15-s + (−0.0871 + 0.268i)16-s + (−3.58 − 2.60i)17-s + ⋯
L(s)  = 1  + (0.611 − 0.198i)2-s + (0.760 + 0.649i)3-s + (−0.474 + 0.344i)4-s − 1.32i·5-s + (0.594 + 0.246i)6-s + (0.0811 − 0.0589i)7-s + (−0.599 + 0.825i)8-s + (0.156 + 0.987i)9-s + (−0.263 − 0.809i)10-s + (−0.210 + 0.152i)11-s + (−0.584 − 0.0458i)12-s + (−0.765 − 0.248i)13-s + (0.0379 − 0.0522i)14-s + (0.859 − 1.00i)15-s + (−0.0217 + 0.0670i)16-s + (−0.869 − 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.994 - 0.106i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.994 - 0.106i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37182 + 0.0730602i\)
\(L(\frac12)\) \(\approx\) \(1.37182 + 0.0730602i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.31 - 1.12i)T \)
31 \( 1 + (-3.83 - 4.03i)T \)
good2 \( 1 + (-0.865 + 0.281i)T + (1.61 - 1.17i)T^{2} \)
5 \( 1 + 2.95iT - 5T^{2} \)
7 \( 1 + (-0.214 + 0.156i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (0.696 - 0.506i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (2.76 + 0.897i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (3.58 + 2.60i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.92 + 5.91i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-7.04 - 5.11i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.42 - 4.38i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 0.956iT - 37T^{2} \)
41 \( 1 + (-1.98 + 0.644i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (-5.22 + 1.69i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-3.72 - 1.20i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.51 - 1.10i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (14.2 + 4.63i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 - 3.93iT - 61T^{2} \)
67 \( 1 + 3.46T + 67T^{2} \)
71 \( 1 + (-6.57 + 9.04i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (8.04 + 11.0i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.62 + 6.36i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.07 + 3.30i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-9.07 + 6.59i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (14.4 - 10.4i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.78614251776810550196200069176, −13.16768917862169944897077379395, −12.31081874774605662489976909116, −10.92842792755305116744762619692, −9.152677356719342822360499799316, −8.977965517652166485762999713344, −7.57530768455856453560571281757, −4.95503875259412887775047096745, −4.67402186519572205028321534565, −2.89658397752766033031698992108, 2.64755631597743976722723158586, 4.15553125179866096117913839315, 6.11943784650201283192656091509, 6.96629447233960417848024730026, 8.348544073833206314807188249908, 9.658768767902066227208426791001, 10.75066005252041433267652595941, 12.29171937339985959940874807135, 13.21936089883724277553299098837, 14.19320017940047639217322881420

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