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Properties

Label 2-57-57.2-c1-0-1
Degree $2$
Conductor $57$
Sign $0.816 - 0.577i$
Analytic cond. $0.455147$
Root an. cond. $0.674646$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.11 + 1.32i)3-s + (1.87 + 0.684i)4-s + (−0.418 − 0.725i)7-s + (−0.520 − 2.95i)9-s + (−3 + 1.73i)12-s + (−4.62 − 5.51i)13-s + (3.06 + 2.57i)16-s + (0.5 + 4.33i)19-s + (1.42 + 0.251i)21-s + (3.83 − 3.21i)25-s + (4.5 + 2.59i)27-s + (−0.290 − 1.64i)28-s + (−4.97 + 2.87i)31-s + (1.04 − 5.90i)36-s + 8.64i·37-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)3-s + (0.939 + 0.342i)4-s + (−0.158 − 0.274i)7-s + (−0.173 − 0.984i)9-s + (−0.866 + 0.500i)12-s + (−1.28 − 1.52i)13-s + (0.766 + 0.642i)16-s + (0.114 + 0.993i)19-s + (0.311 + 0.0549i)21-s + (0.766 − 0.642i)25-s + (0.866 + 0.499i)27-s + (−0.0549 − 0.311i)28-s + (−0.893 + 0.515i)31-s + (0.173 − 0.984i)36-s + 1.42i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 57,\ (\ :1/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.789271 + 0.250821i\)
\(L(\frac12)\) \(\approx\) \(0.789271 + 0.250821i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.11 - 1.32i)T \)
19 \( 1 + (-0.5 - 4.33i)T \)
good2 \( 1 + (-1.87 - 0.684i)T^{2} \)
5 \( 1 + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (0.418 + 0.725i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.62 + 5.51i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (4.97 - 2.87i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.64iT - 37T^{2} \)
41 \( 1 + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (10.9 - 3.97i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (10.1 + 3.68i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-12.7 + 2.24i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-10.8 - 9.08i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-10.1 + 12.1i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-5.11 - 0.902i)T + (91.1 + 33.1i)T^{2} \)
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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

−15.42054957203350118557830031012, −14.67548588204780780464453963076, −12.71095010807082979097509326959, −11.93642718408420124492950504966, −10.64785553138030098936706882748, −9.958638761630887302068269861649, −8.032380330471108314647794843900, −6.63495525779741110114564311192, −5.20493080529604940691327026659, −3.26047994195234543095404170926, 2.20164391801485012388063492124, 5.18183008782740260463291089300, 6.63803538154464438987919619282, 7.37409331340862537581677804791, 9.342745582454400850571802189003, 10.88098729895574099831173953932, 11.72726210824352221627130035698, 12.60762165414567962380544785943, 14.03536306964162902402407385158, 15.15743972780314869660263456988

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