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Properties

Label 2-57-19.12-c2-0-0
Degree $2$
Conductor $57$
Sign $0.263 - 0.964i$
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  + (0.204 − 0.117i)2-s + (−1.5 + 0.866i)3-s + (−1.97 + 3.41i)4-s + (2.88 + 4.98i)5-s + (−0.204 + 0.353i)6-s + 1.94·7-s + 1.87i·8-s + (1.5 − 2.59i)9-s + (1.17 + 0.678i)10-s + 8.46·11-s − 6.83i·12-s + (−16.7 − 9.66i)13-s + (0.396 − 0.229i)14-s + (−8.64 − 4.98i)15-s + (−7.66 − 13.2i)16-s + (12.5 + 21.7i)17-s + ⋯
L(s)  = 1  + (0.102 − 0.0588i)2-s + (−0.5 + 0.288i)3-s + (−0.493 + 0.854i)4-s + (0.576 + 0.997i)5-s + (−0.0340 + 0.0588i)6-s + 0.277·7-s + 0.233i·8-s + (0.166 − 0.288i)9-s + (0.117 + 0.0678i)10-s + 0.769·11-s − 0.569i·12-s + (−1.28 − 0.743i)13-s + (0.0283 − 0.0163i)14-s + (−0.576 − 0.332i)15-s + (−0.479 − 0.830i)16-s + (0.737 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $0.263 - 0.964i$
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: \((2,\ 57,\ (\ :1),\ 0.263 - 0.964i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.837422 + 0.639174i\)
\(L(\frac12)\) \(\approx\) \(0.837422 + 0.639174i\)
\(L(2)\) 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 \( 1 + (-17.8 + 6.59i)T \)
good2 \( 1 + (-0.204 + 0.117i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (-2.88 - 4.98i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 1.94T + 49T^{2} \)
11 \( 1 - 8.46T + 121T^{2} \)
13 \( 1 + (16.7 + 9.66i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-12.5 - 21.7i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-15.6 + 27.0i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-13.7 - 7.91i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 14.4iT - 961T^{2} \)
37 \( 1 - 41.6iT - 1.36e3T^{2} \)
41 \( 1 + (-1.53 + 0.885i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-14.4 - 25.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-1.70 + 2.95i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (80.0 + 46.2i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-4.27 + 2.46i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-7.45 + 12.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (70.4 + 40.6i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-52.9 + 30.5i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (38.8 + 67.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-84.0 + 48.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 102.T + 6.88e3T^{2} \)
89 \( 1 + (-103. - 59.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (42.1 - 24.3i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−14.91627824905453315472698345242, −14.26001970780144610019342551761, −12.82958135686975264968860122733, −11.88281608663695761077599591634, −10.58047301966104785539598470219, −9.539059375677705204092427973476, −7.917144563276463251773311126249, −6.52947701569784611419199066615, −4.86563072204190869620644816458, −3.10020345888458115510291274716, 1.27697268716449623729434958681, 4.78597676192681787709192215491, 5.57447388870786483395014204481, 7.22402978253919709120349939908, 9.206793758987720427240938496625, 9.756379429071805390997627098732, 11.49658653910621823341010138284, 12.48595183723041821046314398519, 13.79457969220269849774722166863, 14.40721522711763656291005847411

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