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Properties

Label 2-57-19.8-c2-0-0
Degree $2$
Conductor $57$
Sign $-0.649 - 0.760i$
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  + (−1.78 − 1.03i)2-s + (−1.5 − 0.866i)3-s + (0.131 + 0.227i)4-s + (−3.20 + 5.55i)5-s + (1.78 + 3.09i)6-s − 2.26·7-s + 7.71i·8-s + (1.5 + 2.59i)9-s + (11.4 − 6.62i)10-s − 20.0·11-s − 0.455i·12-s + (−0.135 + 0.0782i)13-s + (4.04 + 2.33i)14-s + (9.62 − 5.55i)15-s + (8.49 − 14.7i)16-s + (12.3 − 21.3i)17-s + ⋯
L(s)  = 1  + (−0.894 − 0.516i)2-s + (−0.5 − 0.288i)3-s + (0.0328 + 0.0569i)4-s + (−0.641 + 1.11i)5-s + (0.298 + 0.516i)6-s − 0.323·7-s + 0.964i·8-s + (0.166 + 0.288i)9-s + (1.14 − 0.662i)10-s − 1.82·11-s − 0.0379i·12-s + (−0.0104 + 0.00601i)13-s + (0.289 + 0.166i)14-s + (0.641 − 0.370i)15-s + (0.530 − 0.919i)16-s + (0.724 − 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $-0.649 - 0.760i$
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: \((2,\ 57,\ (\ :1),\ -0.649 - 0.760i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0332266 + 0.0720214i\)
\(L(\frac12)\) \(\approx\) \(0.0332266 + 0.0720214i\)
\(L(2)\) 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 \( 1 + (18.9 - 1.60i)T \)
good2 \( 1 + (1.78 + 1.03i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (3.20 - 5.55i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + 2.26T + 49T^{2} \)
11 \( 1 + 20.0T + 121T^{2} \)
13 \( 1 + (0.135 - 0.0782i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-12.3 + 21.3i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (2.62 + 4.54i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (31.4 - 18.1i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 17.1iT - 961T^{2} \)
37 \( 1 - 42.7iT - 1.36e3T^{2} \)
41 \( 1 + (-30.0 - 17.3i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-12.5 + 21.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (14.6 + 25.4i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-48.4 + 27.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (29.9 + 17.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-27.3 - 47.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-66.0 + 38.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-63.6 - 36.7i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (45.9 - 79.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (53.1 + 30.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 148.T + 6.88e3T^{2} \)
89 \( 1 + (62.7 - 36.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (70.0 + 40.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−15.48265905436015998127937803649, −14.33678836936654736615893385511, −12.92570746916365615034336812791, −11.52143980911028228924959882966, −10.72227565355269025859654382249, −9.934783863242824841699040896285, −8.176390519637391945755430688798, −7.11290330523685174443410523990, −5.33322032445728070169032650509, −2.74829494438792967821679412498, 0.10309517159171619023035595239, 4.13649794607816133337634663327, 5.76153876022377098222615039401, 7.65254742692637118163663675371, 8.410677293116563631383377751755, 9.701980947595646565980687348219, 10.83499666808880295452427963251, 12.62915978262048466379473940509, 12.91032614891161551697392314389, 15.22986096867202951223312196463

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