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Properties

Label 2-57-19.12-c2-0-1
Degree $2$
Conductor $57$
Sign $0.762 - 0.646i$
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  + (−0.583 + 0.336i)2-s + (1.5 − 0.866i)3-s + (−1.77 + 3.07i)4-s + (2.27 + 3.93i)5-s + (−0.583 + 1.01i)6-s + 9.87·7-s − 5.08i·8-s + (1.5 − 2.59i)9-s + (−2.65 − 1.53i)10-s − 15.6·11-s + 6.14i·12-s + (13.3 + 7.68i)13-s + (−5.76 + 3.32i)14-s + (6.81 + 3.93i)15-s + (−5.37 − 9.31i)16-s + (−12.4 − 21.5i)17-s + ⋯
L(s)  = 1  + (−0.291 + 0.168i)2-s + (0.5 − 0.288i)3-s + (−0.443 + 0.767i)4-s + (0.454 + 0.787i)5-s + (−0.0972 + 0.168i)6-s + 1.41·7-s − 0.635i·8-s + (0.166 − 0.288i)9-s + (−0.265 − 0.153i)10-s − 1.42·11-s + 0.511i·12-s + (1.02 + 0.590i)13-s + (−0.411 + 0.237i)14-s + (0.454 + 0.262i)15-s + (−0.336 − 0.582i)16-s + (−0.730 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $0.762 - 0.646i$
Analytic conductor: \(1.55313\)
Root analytic conductor: \(1.24624\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 57,\ (\ :1),\ 0.762 - 0.646i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.12568 + 0.412743i\)
\(L(\frac12)\) \(\approx\) \(1.12568 + 0.412743i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 0.866i)T \)
19 \( 1 + (18.4 + 4.63i)T \)
good2 \( 1 + (0.583 - 0.336i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (-2.27 - 3.93i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 9.87T + 49T^{2} \)
11 \( 1 + 15.6T + 121T^{2} \)
13 \( 1 + (-13.3 - 7.68i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (12.4 + 21.5i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-4.15 + 7.19i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-27.3 - 15.7i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 30.9iT - 961T^{2} \)
37 \( 1 - 17.3iT - 1.36e3T^{2} \)
41 \( 1 + (44.2 - 25.5i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (0.773 + 1.33i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-5.09 + 8.81i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-4.54 - 2.62i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-68.4 + 39.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (53.9 - 93.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (16.9 + 9.80i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (45.3 - 26.1i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (36.6 + 63.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (27.1 - 15.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 42.6T + 6.88e3T^{2} \)
89 \( 1 + (-114. - 66.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (70.2 - 40.5i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−14.99358649455165137860582076299, −13.83237713801458074459458981059, −13.23197158296259952468761438598, −11.61226371295480783868485332865, −10.47115744817376458762178731455, −8.829518203180183606310686427972, −8.019673128765394307121144189346, −6.81071120763688348567476280827, −4.64476772033538775714580879762, −2.59285601662727356769872241821, 1.72904997283136517504779659648, 4.61545037830337431409523359732, 5.63808262593136137713105384758, 8.304772319891495686755460089446, 8.642381471455556935974416271686, 10.33345326715855229218692463591, 10.91290810091182261389768669465, 12.92185015062862269761323362569, 13.72323026732987238603410592984, 14.89039502146008261830912679069

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