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Properties

Label 2-57-19.18-c2-0-5
Degree $2$
Conductor $57$
Sign $-0.917 + 0.397i$
Analytic cond. $1.55313$
Root an. cond. $1.24624$
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  − 3.04i·2-s − 1.73i·3-s − 5.27·4-s − 6.27·5-s − 5.27·6-s + 12.2·7-s + 3.88i·8-s − 2.99·9-s + 19.1i·10-s + 0.274·11-s + 9.13i·12-s − 13.0i·13-s − 37.3i·14-s + 10.8i·15-s − 9.27·16-s + 17.3·17-s + ⋯
L(s)  = 1  − 1.52i·2-s − 0.577i·3-s − 1.31·4-s − 1.25·5-s − 0.879·6-s + 1.75·7-s + 0.485i·8-s − 0.333·9-s + 1.91i·10-s + 0.0249·11-s + 0.761i·12-s − 1.00i·13-s − 2.67i·14-s + 0.724i·15-s − 0.579·16-s + 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $-0.917 + 0.397i$
Analytic conductor: \(1.55313\)
Root analytic conductor: \(1.24624\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 57,\ (\ :1),\ -0.917 + 0.397i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.215995 - 1.04239i\)
\(L(\frac12)\) \(\approx\) \(0.215995 - 1.04239i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
19 \( 1 + (-7.54 - 17.4i)T \)
good2 \( 1 + 3.04iT - 4T^{2} \)
5 \( 1 + 6.27T + 25T^{2} \)
7 \( 1 - 12.2T + 49T^{2} \)
11 \( 1 - 0.274T + 121T^{2} \)
13 \( 1 + 13.0iT - 169T^{2} \)
17 \( 1 - 17.3T + 289T^{2} \)
23 \( 1 - 20.5T + 529T^{2} \)
29 \( 1 - 26.0iT - 841T^{2} \)
31 \( 1 + 23.5iT - 961T^{2} \)
37 \( 1 - 66.7iT - 1.36e3T^{2} \)
41 \( 1 - 3.57iT - 1.68e3T^{2} \)
43 \( 1 + 48.1T + 1.84e3T^{2} \)
47 \( 1 + 12.4T + 2.20e3T^{2} \)
53 \( 1 + 25.8iT - 2.80e3T^{2} \)
59 \( 1 - 0.230iT - 3.48e3T^{2} \)
61 \( 1 + 28.8T + 3.72e3T^{2} \)
67 \( 1 + 102. iT - 4.48e3T^{2} \)
71 \( 1 - 107. iT - 5.04e3T^{2} \)
73 \( 1 - 11.1T + 5.32e3T^{2} \)
79 \( 1 + 26.6iT - 6.24e3T^{2} \)
83 \( 1 + 89.6T + 6.88e3T^{2} \)
89 \( 1 + 139. iT - 7.92e3T^{2} \)
97 \( 1 + 41.3iT - 9.40e3T^{2} \)
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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

−14.29773330429146035907241574058, −12.90027712548143265349839351515, −11.85045020263953769238325726810, −11.43751161932250064193366607055, −10.30959350061867051072233102905, −8.393218647446738802228616811803, −7.62348773550942138717445594476, −4.95234041931024610192874865350, −3.35334285955497252505292511115, −1.27946632850648438308219700368, 4.29938861470364890416464825220, 5.24489624718865330608606822678, 7.19656061476251659370600715079, 8.031676185563881664076279202167, 8.999869473752230897751361533189, 11.09344555195370457874892071452, 11.80692615019403898550036843074, 13.96540824083863286578772865298, 14.72300263131676798033107809626, 15.40734100412113252571816446618

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