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Properties

Label 2-57-57.26-c2-0-0
Degree $2$
Conductor $57$
Sign $-0.924 + 0.382i$
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  + (−2.00 + 1.15i)2-s + (−1.67 + 2.48i)3-s + (0.682 − 1.18i)4-s + (2.24 − 1.29i)5-s + (0.484 − 6.93i)6-s − 12.9·7-s − 6.10i·8-s + (−3.37 − 8.34i)9-s + (−3.00 + 5.20i)10-s + 14.5i·11-s + (1.79 + 3.68i)12-s + (2.52 − 4.36i)13-s + (26.0 − 15.0i)14-s + (−0.542 + 7.75i)15-s + (9.79 + 16.9i)16-s + (−15.9 + 9.19i)17-s + ⋯
L(s)  = 1  + (−1.00 + 0.579i)2-s + (−0.559 + 0.829i)3-s + (0.170 − 0.295i)4-s + (0.449 − 0.259i)5-s + (0.0807 − 1.15i)6-s − 1.85·7-s − 0.762i·8-s + (−0.374 − 0.927i)9-s + (−0.300 + 0.520i)10-s + 1.31i·11-s + (0.149 + 0.306i)12-s + (0.193 − 0.335i)13-s + (1.85 − 1.07i)14-s + (−0.0361 + 0.517i)15-s + (0.612 + 1.06i)16-s + (−0.936 + 0.540i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.382i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.924 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $-0.924 + 0.382i$
Analytic conductor: \(1.55313\)
Root analytic conductor: \(1.24624\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 57,\ (\ :1),\ -0.924 + 0.382i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0479803 - 0.241511i\)
\(L(\frac12)\) \(\approx\) \(0.0479803 - 0.241511i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 - 2.48i)T \)
19 \( 1 + (-2.86 - 18.7i)T \)
good2 \( 1 + (2.00 - 1.15i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (-2.24 + 1.29i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + 12.9T + 49T^{2} \)
11 \( 1 - 14.5iT - 121T^{2} \)
13 \( 1 + (-2.52 + 4.36i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (15.9 - 9.19i)T + (144.5 - 250. i)T^{2} \)
23 \( 1 + (8.57 + 4.94i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-7.48 - 4.32i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 13.3T + 961T^{2} \)
37 \( 1 - 7.28T + 1.36e3T^{2} \)
41 \( 1 + (11.5 - 6.64i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (19.6 + 34.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-13.5 - 7.84i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-41.6 - 24.0i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-64.8 + 37.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (5.14 - 8.91i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (39.7 - 68.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-9.59 + 5.53i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (4.59 + 7.95i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (54.2 + 93.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 102. iT - 6.88e3T^{2} \)
89 \( 1 + (81.8 + 47.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-21.4 - 37.1i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−15.89626125375777022586736599240, −15.15221146529890950913088111893, −13.12813845722215557693829550076, −12.33970156924161299434000723636, −10.26467170198890982955030910490, −9.792495741857297842196757826459, −8.866827269005312880690855612711, −6.98552687564934913656820875166, −5.94473501327457447465665484456, −3.85692897879068622863265159102, 0.33237159918131911688646251845, 2.67238256188768911199961885495, 5.84618833800110864944915724580, 6.84074508634642613783830340620, 8.638237237656222694337479822392, 9.694334711549750210465036631034, 10.81754142750205109602338660893, 11.77821667600917672823522843632, 13.25097687064847172199878901051, 13.82635301731285998901669604502

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