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Properties

Label 2-57-19.8-c2-0-5
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: $\chi_{57} (46, \cdot )$
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

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

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