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Properties

Label 2-93-93.53-c1-0-7
Degree $2$
Conductor $93$
Sign $0.311 + 0.950i$
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  + (0.360 − 1.69i)3-s + (−1.61 − 1.17i)4-s + (1.88 − 0.837i)7-s + (−2.74 − 1.22i)9-s + (−2.57 + 2.31i)12-s + (5.30 + 4.77i)13-s + (1.23 + 3.80i)16-s + (−1.74 − 1.93i)19-s + (−0.741 − 3.48i)21-s + (−2.5 + 4.33i)25-s + (−3.05 + 4.20i)27-s + (−4.02 − 0.855i)28-s + (5.55 + 0.296i)31-s + (3.00 + 5.19i)36-s + (−10.4 − 6.03i)37-s + ⋯
L(s)  = 1  + (0.207 − 0.978i)3-s + (−0.809 − 0.587i)4-s + (0.710 − 0.316i)7-s + (−0.913 − 0.406i)9-s + (−0.743 + 0.669i)12-s + (1.47 + 1.32i)13-s + (0.309 + 0.951i)16-s + (−0.400 − 0.444i)19-s + (−0.161 − 0.761i)21-s + (−0.5 + 0.866i)25-s + (−0.587 + 0.809i)27-s + (−0.761 − 0.161i)28-s + (0.998 + 0.0532i)31-s + (0.500 + 0.866i)36-s + (−1.71 − 0.991i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.311 + 0.950i$
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: \((2,\ 93,\ (\ :1/2),\ 0.311 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.767136 - 0.556053i\)
\(L(\frac12)\) \(\approx\) \(0.767136 - 0.556053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.360 + 1.69i)T \)
31 \( 1 + (-5.55 - 0.296i)T \)
good2 \( 1 + (1.61 + 1.17i)T^{2} \)
5 \( 1 + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.88 + 0.837i)T + (4.68 - 5.20i)T^{2} \)
11 \( 1 + (-10.7 - 2.28i)T^{2} \)
13 \( 1 + (-5.30 - 4.77i)T + (1.35 + 12.9i)T^{2} \)
17 \( 1 + (-16.6 + 3.53i)T^{2} \)
19 \( 1 + (1.74 + 1.93i)T + (-1.98 + 18.8i)T^{2} \)
23 \( 1 + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + (10.4 + 6.03i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-37.4 + 16.6i)T^{2} \)
43 \( 1 + (-6.31 + 5.68i)T + (4.49 - 42.7i)T^{2} \)
47 \( 1 + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (35.4 + 39.3i)T^{2} \)
59 \( 1 + (-53.8 - 23.9i)T^{2} \)
61 \( 1 - 2.62iT - 61T^{2} \)
67 \( 1 + (-7.49 - 12.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-47.5 - 52.7i)T^{2} \)
73 \( 1 + (15.5 + 1.63i)T + (71.4 + 15.1i)T^{2} \)
79 \( 1 + (11.5 - 1.21i)T + (77.2 - 16.4i)T^{2} \)
83 \( 1 + (75.8 - 33.7i)T^{2} \)
89 \( 1 + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (5.78 + 4.20i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.85458818721137333825887100264, −13.10419794021573190048235161162, −11.70201205416395250082268342010, −10.75965427661621037297515438020, −9.115176909062379307616114058469, −8.417789438325054974218084501880, −6.95902906108432179939819737123, −5.70979325045132626554403560203, −4.08730349124566849199642013513, −1.52666541951277348169336733726, 3.28446896734065664730281152784, 4.53150263395175964780550367247, 5.76574762956726733518220062836, 8.197410635356018385709191441887, 8.481962763040090094617800351645, 9.899129624581495638820549813985, 10.90507135998168467913405538257, 12.12295591183965126138618775269, 13.38993659570027584139756546937, 14.22393300214343963544804096237

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