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Properties

Label 2-93-93.83-c1-0-4
Degree $2$
Conductor $93$
Sign $0.965 - 0.260i$
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  + (1.72 − 0.181i)3-s + (0.618 + 1.90i)4-s + (−5.15 − 1.09i)7-s + (2.93 − 0.623i)9-s + (1.40 + 3.16i)12-s + (2.58 − 5.81i)13-s + (−3.23 + 2.35i)16-s + (−2.53 + 1.13i)19-s + (−9.08 − 0.954i)21-s + (−2.5 + 4.33i)25-s + (4.94 − 1.60i)27-s + (−1.10 − 10.4i)28-s + (1.43 + 5.37i)31-s + (3.00 + 5.19i)36-s + (−1.95 − 1.13i)37-s + ⋯
L(s)  = 1  + (0.994 − 0.104i)3-s + (0.309 + 0.951i)4-s + (−1.94 − 0.414i)7-s + (0.978 − 0.207i)9-s + (0.406 + 0.913i)12-s + (0.717 − 1.61i)13-s + (−0.809 + 0.587i)16-s + (−0.582 + 0.259i)19-s + (−1.98 − 0.208i)21-s + (−0.5 + 0.866i)25-s + (0.951 − 0.309i)27-s + (−0.208 − 1.98i)28-s + (0.257 + 0.966i)31-s + (0.500 + 0.866i)36-s + (−0.322 − 0.185i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $0.965 - 0.260i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ 0.965 - 0.260i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19998 + 0.159221i\)
\(L(\frac12)\) \(\approx\) \(1.19998 + 0.159221i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 + 0.181i)T \)
31 \( 1 + (-1.43 - 5.37i)T \)
good2 \( 1 + (-0.618 - 1.90i)T^{2} \)
5 \( 1 + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (5.15 + 1.09i)T + (6.39 + 2.84i)T^{2} \)
11 \( 1 + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (-2.58 + 5.81i)T + (-8.69 - 9.66i)T^{2} \)
17 \( 1 + (-1.77 + 16.9i)T^{2} \)
19 \( 1 + (2.53 - 1.13i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (8.96 + 27.5i)T^{2} \)
37 \( 1 + (1.95 + 1.13i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (40.1 + 8.52i)T^{2} \)
43 \( 1 + (-2.79 - 6.27i)T + (-28.7 + 31.9i)T^{2} \)
47 \( 1 + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (48.4 - 21.5i)T^{2} \)
59 \( 1 + (57.7 - 12.2i)T^{2} \)
61 \( 1 + 11.1iT - 61T^{2} \)
67 \( 1 + (4.11 + 7.13i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-64.8 + 28.8i)T^{2} \)
73 \( 1 + (-8.46 - 7.62i)T + (7.63 + 72.6i)T^{2} \)
79 \( 1 + (1.15 - 1.03i)T + (8.25 - 78.5i)T^{2} \)
83 \( 1 + (-81.1 - 17.2i)T^{2} \)
89 \( 1 + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.12 + 15.7i)T + (-78.4 + 57.0i)T^{2} \)
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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

−13.76404902871764960432637772034, −12.89687942145821819202868881338, −12.58842822004627534368769105443, −10.67447261232272083154635181993, −9.619178243383604244398354422838, −8.467147877568407880838916901335, −7.41063610732153479335109726691, −6.32936944027612822576010998756, −3.69064321658200426380421896390, −3.01025219821789969513365137599, 2.38344690633852038406427839166, 4.02817152785137470180630507046, 6.14404177466395507931258090812, 6.90971675921872657199118631133, 8.871523212563062389952183815540, 9.529721708604119078688739492254, 10.42685596701800289772962328269, 11.96552956505323863583632023220, 13.27112219263887555300969974121, 13.92755983252103761784140582325

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