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Properties

Label 2-57-19.18-c2-0-2
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  + 1.31i·2-s − 1.73i·3-s + 2.27·4-s + 1.27·5-s + 2.27·6-s + 4.72·7-s + 8.24i·8-s − 2.99·9-s + 1.67i·10-s − 7.27·11-s − 3.94i·12-s − 4.30i·13-s + 6.20i·14-s − 2.20i·15-s − 1.72·16-s − 20.3·17-s + ⋯
L(s)  = 1  + 0.656i·2-s − 0.577i·3-s + 0.568·4-s + 0.254·5-s + 0.379·6-s + 0.675·7-s + 1.03i·8-s − 0.333·9-s + 0.167i·10-s − 0.661·11-s − 0.328i·12-s − 0.330i·13-s + 0.443i·14-s − 0.147i·15-s − 0.107·16-s − 1.19·17-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

\(L(\frac{3}{2})\) \(\approx\) \(1.34137 + 0.277947i\)
\(L(\frac12)\) \(\approx\) \(1.34137 + 0.277947i\)
\(L(2)\) 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 \( 1 - 1.31iT - 4T^{2} \)
5 \( 1 - 1.27T + 25T^{2} \)
7 \( 1 - 4.72T + 49T^{2} \)
11 \( 1 + 7.27T + 121T^{2} \)
13 \( 1 + 4.30iT - 169T^{2} \)
17 \( 1 + 20.3T + 289T^{2} \)
23 \( 1 - 5.45T + 529T^{2} \)
29 \( 1 - 8.60iT - 841T^{2} \)
31 \( 1 - 20.0iT - 961T^{2} \)
37 \( 1 - 40.6iT - 1.36e3T^{2} \)
41 \( 1 + 31.2iT - 1.68e3T^{2} \)
43 \( 1 - 65.1T + 1.84e3T^{2} \)
47 \( 1 - 55.4T + 2.20e3T^{2} \)
53 \( 1 + 78.1iT - 2.80e3T^{2} \)
59 \( 1 + 69.5iT - 3.48e3T^{2} \)
61 \( 1 + 6.17T + 3.72e3T^{2} \)
67 \( 1 - 123. iT - 4.48e3T^{2} \)
71 \( 1 - 3.11iT - 5.04e3T^{2} \)
73 \( 1 - 33.8T + 5.32e3T^{2} \)
79 \( 1 + 87.6iT - 6.24e3T^{2} \)
83 \( 1 - 121.T + 6.88e3T^{2} \)
89 \( 1 - 69.9iT - 7.92e3T^{2} \)
97 \( 1 + 111. iT - 9.40e3T^{2} \)
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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

−15.17871534424756598897372666555, −14.03959812627689957020290602945, −12.96594384674733419137649408753, −11.56761066480337137135847442264, −10.67291369081433945522225662846, −8.704806974371689093596230835705, −7.61872590906246447706973124949, −6.51067166346380568734738365123, −5.15014398247745938080575999639, −2.29933219292304627208110346809, 2.25782140772410595349040478687, 4.19925424629104084394854135940, 5.99590177530825523476550265131, 7.70515736440860893674332700935, 9.299896432692565940943930923427, 10.54944133575905060915345133821, 11.23497522896712232025697974523, 12.42770639383669636760763477144, 13.73182479112134989666089951235, 15.05715978563358812406575014526

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