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Properties

Label 2-93-93.92-c1-0-1
Degree $2$
Conductor $93$
Sign $0.937 - 0.349i$
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  − 0.834i·2-s + (−0.590 + 1.62i)3-s + 1.30·4-s + 2.75i·5-s + (1.35 + 0.493i)6-s − 7-s − 2.75i·8-s + (−2.30 − 1.92i)9-s + 2.30·10-s + 5.08·11-s + (−0.769 + 2.12i)12-s − 4.24i·13-s + 0.834i·14-s + (−4.49 − 1.62i)15-s + 0.302·16-s − 6.61·17-s + ⋯
L(s)  = 1  − 0.590i·2-s + (−0.340 + 0.940i)3-s + 0.651·4-s + 1.23i·5-s + (0.555 + 0.201i)6-s − 0.377·7-s − 0.975i·8-s + (−0.767 − 0.640i)9-s + 0.728·10-s + 1.53·11-s + (−0.222 + 0.612i)12-s − 1.17i·13-s + 0.223i·14-s + (−1.15 − 0.420i)15-s + 0.0756·16-s − 1.60·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.937 - 0.349i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.937 - 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00240 + 0.180614i\)
\(L(\frac12)\) \(\approx\) \(1.00240 + 0.180614i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.590 - 1.62i)T \)
31 \( 1 + (-3.60 - 4.24i)T \)
good2 \( 1 + 0.834iT - 2T^{2} \)
5 \( 1 - 2.75iT - 5T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 - 5.08T + 11T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 + 6.61T + 17T^{2} \)
19 \( 1 + 3.60T + 19T^{2} \)
23 \( 1 - 1.53T + 23T^{2} \)
29 \( 1 + 1.53T + 29T^{2} \)
37 \( 1 + 5.52iT - 37T^{2} \)
41 \( 1 + 6.60iT - 41T^{2} \)
43 \( 1 - 4.24iT - 43T^{2} \)
47 \( 1 - 3.84iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 2.75iT - 59T^{2} \)
61 \( 1 - 9.76iT - 61T^{2} \)
67 \( 1 + 0.605T + 67T^{2} \)
71 \( 1 - 4.93iT - 71T^{2} \)
73 \( 1 - 5.52iT - 73T^{2} \)
79 \( 1 + 1.28iT - 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 6.61T + 89T^{2} \)
97 \( 1 + 8.81T + 97T^{2} \)
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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

−14.40927601474074427588679529679, −12.77658601154242751535836240248, −11.56163991811995277235837187328, −10.87304918635273789950780092432, −10.20179122382072860034461461582, −8.983026135629515160336995810595, −6.90005381522343535699841591011, −6.18217096370193385550249085774, −3.99233210079300137381162760878, −2.83574624922866181894535576710, 1.81554026401557863223141292888, 4.62680997506675995294583726357, 6.32496316493280144748409313064, 6.77170210626133481199784310036, 8.345658300129451251005764616573, 9.166958158817773083946440367850, 11.25713770450423635949160663517, 11.87374316248355948584226391111, 12.90856406756892673439493494488, 13.88844367710648568161544535322

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