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Properties

Label 2-93-93.26-c1-0-6
Degree $2$
Conductor $93$
Sign $0.345 + 0.938i$
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  − 0.792i·2-s + (0.5 − 1.65i)3-s + 1.37·4-s + (−0.686 + 0.396i)5-s + (−1.31 − 0.396i)6-s + (−1.68 + 2.92i)7-s − 2.67i·8-s + (−2.5 − 1.65i)9-s + (0.313 + 0.543i)10-s + (0.686 + 1.18i)11-s + (0.686 − 2.27i)12-s + (1.5 − 0.866i)13-s + (2.31 + 1.33i)14-s + (0.313 + 1.33i)15-s + 0.627·16-s + (−3.68 + 6.38i)17-s + ⋯
L(s)  = 1  − 0.560i·2-s + (0.288 − 0.957i)3-s + 0.686·4-s + (−0.306 + 0.177i)5-s + (−0.536 − 0.161i)6-s + (−0.637 + 1.10i)7-s − 0.944i·8-s + (−0.833 − 0.552i)9-s + (0.0992 + 0.171i)10-s + (0.206 + 0.358i)11-s + (0.198 − 0.656i)12-s + (0.416 − 0.240i)13-s + (0.618 + 0.357i)14-s + (0.0810 + 0.344i)15-s + 0.156·16-s + (−0.894 + 1.54i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.345 + 0.938i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.345 + 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.920992 - 0.642031i\)
\(L(\frac12)\) \(\approx\) \(0.920992 - 0.642031i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 1.65i)T \)
31 \( 1 + (2 + 5.19i)T \)
good2 \( 1 + 0.792iT - 2T^{2} \)
5 \( 1 + (0.686 - 0.396i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.68 - 2.92i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.686 - 1.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.5 + 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.68 - 6.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.87 + 3.24i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
37 \( 1 + (-1.5 - 0.866i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.686 - 0.396i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.5 + 4.33i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.63iT - 47T^{2} \)
53 \( 1 + (-0.686 - 1.18i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (12.4 + 7.17i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (3.05 + 5.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-12.4 + 7.17i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (11.6 - 6.70i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.05 - 4.65i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.68 - 11.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 0.372T + 97T^{2} \)
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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

−13.44188537439652062159680192763, −12.61811824931353779717163159209, −11.87685276066619699144274451069, −10.93314172273215649297787318362, −9.432255554250412209894507191548, −8.244014882442956802785771458872, −6.86745120774974781890472243378, −6.03370231387674633723285728219, −3.38041093367723639633173456768, −2.06928545456695564652782381382, 3.17382139214455280238071597938, 4.63276108881300892416901041928, 6.26687808977160168981242657698, 7.44213978080401157631925179599, 8.659542931284070488183857090626, 9.955001251909501017892956685447, 10.94143537480813422143196738544, 11.87466485184409989130922634845, 13.67772965736870454209641914373, 14.27121817707137232461324207991

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