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Properties

Label 2-93-31.4-c1-0-3
Degree $2$
Conductor $93$
Sign $0.999 - 0.0274i$
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  + (0.239 + 0.735i)2-s + (0.309 − 0.951i)3-s + (1.13 − 0.823i)4-s − 1.00·5-s + 0.773·6-s + (0.312 − 0.226i)7-s + (2.12 + 1.54i)8-s + (−0.809 − 0.587i)9-s + (−0.240 − 0.738i)10-s + (−3.14 + 2.28i)11-s + (−0.433 − 1.33i)12-s + (−0.958 + 2.95i)13-s + (0.241 + 0.175i)14-s + (−0.310 + 0.954i)15-s + (0.236 − 0.729i)16-s + (−0.0512 − 0.0372i)17-s + ⋯
L(s)  = 1  + (0.169 + 0.520i)2-s + (0.178 − 0.549i)3-s + (0.566 − 0.411i)4-s − 0.449·5-s + 0.315·6-s + (0.118 − 0.0857i)7-s + (0.752 + 0.546i)8-s + (−0.269 − 0.195i)9-s + (−0.0759 − 0.233i)10-s + (−0.947 + 0.688i)11-s + (−0.125 − 0.384i)12-s + (−0.265 + 0.818i)13-s + (0.0645 + 0.0469i)14-s + (−0.0801 + 0.246i)15-s + (0.0592 − 0.182i)16-s + (−0.0124 − 0.00902i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0274i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.999 - 0.0274i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.999 - 0.0274i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17183 + 0.0160877i\)
\(L(\frac12)\) \(\approx\) \(1.17183 + 0.0160877i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.362 + 5.55i)T \)
good2 \( 1 + (-0.239 - 0.735i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + 1.00T + 5T^{2} \)
7 \( 1 + (-0.312 + 0.226i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (3.14 - 2.28i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.958 - 2.95i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.0512 + 0.0372i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.514 + 1.58i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.64 + 1.91i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.788 - 2.42i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 7.08T + 37T^{2} \)
41 \( 1 + (-0.240 - 0.738i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-3.55 - 10.9i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-3.63 + 11.1i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.10 - 2.97i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-4.21 + 12.9i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 - 8.74T + 67T^{2} \)
71 \( 1 + (6.78 + 4.93i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.07 + 2.95i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.77 + 2.74i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.29 + 10.1i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (2.09 - 1.52i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (10.7 - 7.78i)T + (29.9 - 92.2i)T^{2} \)
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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

−14.21736165366577043053937175395, −13.12390292795734473471221010652, −11.93193495525807016042883628324, −10.98794941642880240632039526671, −9.701410350392719540648377228481, −8.009021932115818194087776940764, −7.28247273565937376651961176116, −6.10981911767364964321924248775, −4.61042632860311333478667958614, −2.25382051081466301735245451376, 2.72588631374159526754021139107, 3.98354775242182671940521353758, 5.66431474920827766043351699356, 7.51965886991713277969933811903, 8.348962411882504392269406648859, 10.05527771065773544050102393684, 10.86269374579288553463798465945, 11.82913438531817948540650605453, 12.83482011917011039192000973701, 13.91056554583299398680661824571

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