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Properties

Label 2-93-31.6-c2-0-3
Degree $2$
Conductor $93$
Sign $0.988 - 0.152i$
Analytic cond. $2.53406$
Root an. cond. $1.59187$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.62·2-s + (−1.5 + 0.866i)3-s − 1.36·4-s + (2.11 − 3.66i)5-s + (2.43 − 1.40i)6-s + (2.02 + 3.50i)7-s + 8.70·8-s + (1.5 − 2.59i)9-s + (−3.43 + 5.94i)10-s + (16.7 + 9.65i)11-s + (2.04 − 1.18i)12-s + (0.546 + 0.315i)13-s + (−3.28 − 5.69i)14-s + 7.32i·15-s − 8.68·16-s + (19.1 − 11.0i)17-s + ⋯
L(s)  = 1  − 0.811·2-s + (−0.5 + 0.288i)3-s − 0.341·4-s + (0.422 − 0.732i)5-s + (0.405 − 0.234i)6-s + (0.289 + 0.501i)7-s + 1.08·8-s + (0.166 − 0.288i)9-s + (−0.343 + 0.594i)10-s + (1.51 + 0.877i)11-s + (0.170 − 0.0984i)12-s + (0.0420 + 0.0242i)13-s + (−0.234 − 0.406i)14-s + 0.488i·15-s − 0.542·16-s + (1.12 − 0.650i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.988 - 0.152i$
Analytic conductor: \(2.53406\)
Root analytic conductor: \(1.59187\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1),\ 0.988 - 0.152i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.790937 + 0.0608505i\)
\(L(\frac12)\) \(\approx\) \(0.790937 + 0.0608505i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 0.866i)T \)
31 \( 1 + (2.82 - 30.8i)T \)
good2 \( 1 + 1.62T + 4T^{2} \)
5 \( 1 + (-2.11 + 3.66i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-2.02 - 3.50i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-16.7 - 9.65i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-0.546 - 0.315i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-19.1 + 11.0i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (2.54 + 4.40i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + 8.72iT - 529T^{2} \)
29 \( 1 + 23.0iT - 841T^{2} \)
37 \( 1 + (22.9 - 13.2i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (13.9 - 24.0i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-56.0 + 32.3i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 25.0T + 2.20e3T^{2} \)
53 \( 1 + (36.4 + 21.0i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-53.4 - 92.5i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 - 77.5iT - 3.72e3T^{2} \)
67 \( 1 + (-31.4 + 54.4i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-30.2 + 52.4i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (65.5 + 37.8i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (125. - 72.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (86.9 + 50.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 63.6iT - 7.92e3T^{2} \)
97 \( 1 + 83.3T + 9.40e3T^{2} \)
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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

−13.86293127837510873067906860796, −12.52397190643984331178373687183, −11.71303762421121665057695199645, −10.22263740899749537998068645317, −9.364945753184656625938252192447, −8.678120150441318141561725177743, −7.11732340998281745156821702068, −5.44577229063113592547060409579, −4.34066417435096316659365588680, −1.30291935888632651091653464362, 1.23304840527883967994022859777, 3.91470932603744665806942753621, 5.79693657929903183488062387238, 7.02091097682023571535038845291, 8.223611719293924547494890747373, 9.461003925262522351283067572306, 10.49849160756074156464301177448, 11.29380589830251024250693792008, 12.66811043547084284700137338980, 14.09374604332678128585331505147

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