Properties

Label 2-93-93.53-c1-0-1
Degree $2$
Conductor $93$
Sign $0.862 + 0.505i$
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  + (−1.16 − 0.378i)2-s + (−1.32 + 1.11i)3-s + (−0.405 − 0.294i)4-s + (2.19 − 1.26i)5-s + (1.96 − 0.791i)6-s + (4.16 − 1.85i)7-s + (1.79 + 2.47i)8-s + (0.530 − 2.95i)9-s + (−3.03 + 0.645i)10-s + (−0.275 + 2.62i)11-s + (0.865 − 0.0590i)12-s + (−2.55 − 2.29i)13-s + (−5.55 + 0.583i)14-s + (−1.50 + 4.12i)15-s + (−0.848 − 2.61i)16-s + (−0.337 − 3.20i)17-s + ⋯
L(s)  = 1  + (−0.823 − 0.267i)2-s + (−0.767 + 0.641i)3-s + (−0.202 − 0.147i)4-s + (0.982 − 0.567i)5-s + (0.803 − 0.322i)6-s + (1.57 − 0.701i)7-s + (0.636 + 0.875i)8-s + (0.176 − 0.984i)9-s + (−0.960 + 0.204i)10-s + (−0.0831 + 0.791i)11-s + (0.249 − 0.0170i)12-s + (−0.708 − 0.637i)13-s + (−1.48 + 0.156i)14-s + (−0.389 + 1.06i)15-s + (−0.212 − 0.653i)16-s + (−0.0817 − 0.777i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.862 + 0.505i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.862 + 0.505i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.611832 - 0.166058i\)
\(L(\frac12)\) \(\approx\) \(0.611832 - 0.166058i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.32 - 1.11i)T \)
31 \( 1 + (1.29 - 5.41i)T \)
good2 \( 1 + (1.16 + 0.378i)T + (1.61 + 1.17i)T^{2} \)
5 \( 1 + (-2.19 + 1.26i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-4.16 + 1.85i)T + (4.68 - 5.20i)T^{2} \)
11 \( 1 + (0.275 - 2.62i)T + (-10.7 - 2.28i)T^{2} \)
13 \( 1 + (2.55 + 2.29i)T + (1.35 + 12.9i)T^{2} \)
17 \( 1 + (0.337 + 3.20i)T + (-16.6 + 3.53i)T^{2} \)
19 \( 1 + (-1.53 - 1.70i)T + (-1.98 + 18.8i)T^{2} \)
23 \( 1 + (-3.78 + 2.74i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.05 - 6.33i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + (1.86 + 1.07i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.643 + 3.02i)T + (-37.4 + 16.6i)T^{2} \)
43 \( 1 + (3.49 - 3.14i)T + (4.49 - 42.7i)T^{2} \)
47 \( 1 + (9.44 - 3.06i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.98 + 1.32i)T + (35.4 + 39.3i)T^{2} \)
59 \( 1 + (-0.0342 + 0.161i)T + (-53.8 - 23.9i)T^{2} \)
61 \( 1 - 3.09iT - 61T^{2} \)
67 \( 1 + (0.811 + 1.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.95 - 11.1i)T + (-47.5 - 52.7i)T^{2} \)
73 \( 1 + (-7.30 - 0.768i)T + (71.4 + 15.1i)T^{2} \)
79 \( 1 + (8.48 - 0.891i)T + (77.2 - 16.4i)T^{2} \)
83 \( 1 + (-9.39 + 1.99i)T + (75.8 - 33.7i)T^{2} \)
89 \( 1 + (3.71 + 2.70i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (1.66 + 1.21i)T + (29.9 + 92.2i)T^{2} \)
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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

−14.16262541585120968385234883106, −12.76146266709974462888732945102, −11.40838102281490738779002242995, −10.49125845315916649537963962778, −9.795507481583301716578964935373, −8.772410738282401091062324208991, −7.35063866455107756976789672587, −5.13580745036995222874387405883, −4.90552088200411497793600763587, −1.42013379859021656917928206818, 1.88368296663992333412517486743, 4.93844364673481151420535288024, 6.16830100342694073568288087165, 7.49622619650175766305341958446, 8.451648216052377054607882644159, 9.712640361352070858973028073377, 10.95478325593106481462586431708, 11.76164376953878487924986710634, 13.19570694209226560050195842256, 13.98487284275129011724484896614

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