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Properties

Label 2-57-57.56-c1-0-3
Degree $2$
Conductor $57$
Sign $0.943 + 0.332i$
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  + 1.41·2-s + (−0.707 − 1.58i)3-s + 2.23i·5-s + (−1.00 − 2.23i)6-s + 7-s − 2.82·8-s + (−2.00 + 2.23i)9-s + 3.16i·10-s − 2.23i·11-s + 3.16i·13-s + 1.41·14-s + (3.53 − 1.58i)15-s − 4.00·16-s − 6.70i·17-s + (−2.82 + 3.16i)18-s + (3 − 3.16i)19-s + ⋯
L(s)  = 1  + 1.00·2-s + (−0.408 − 0.912i)3-s + 0.999i·5-s + (−0.408 − 0.912i)6-s + 0.377·7-s − 0.999·8-s + (−0.666 + 0.745i)9-s + 1.00i·10-s − 0.674i·11-s + 0.877i·13-s + 0.377·14-s + (0.912 − 0.408i)15-s − 1.00·16-s − 1.62i·17-s + (−0.666 + 0.745i)18-s + (0.688 − 0.725i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $0.943 + 0.332i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 57,\ (\ :1/2),\ 0.943 + 0.332i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07822 - 0.184271i\)
\(L(\frac12)\) \(\approx\) \(1.07822 - 0.184271i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 + 1.58i)T \)
19 \( 1 + (-3 + 3.16i)T \)
good2 \( 1 - 1.41T + 2T^{2} \)
5 \( 1 - 2.23iT - 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 2.23iT - 11T^{2} \)
13 \( 1 - 3.16iT - 13T^{2} \)
17 \( 1 + 6.70iT - 17T^{2} \)
23 \( 1 - 4.47iT - 23T^{2} \)
29 \( 1 + 5.65T + 29T^{2} \)
31 \( 1 + 3.16iT - 31T^{2} \)
37 \( 1 - 9.48iT - 37T^{2} \)
41 \( 1 - 9.89T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 - 2.23iT - 47T^{2} \)
53 \( 1 + 4.24T + 53T^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 - 6.32iT - 67T^{2} \)
71 \( 1 - 4.24T + 71T^{2} \)
73 \( 1 + 3T + 73T^{2} \)
79 \( 1 + 12.6iT - 79T^{2} \)
83 \( 1 + 8.94iT - 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 - 3.16iT - 97T^{2} \)
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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

−14.69119031246940625234437704326, −13.88418300241703166462681872186, −13.27774734279471841401361491382, −11.65809790001365106892915998469, −11.34595879883026444447957287624, −9.269834507866149855137805291709, −7.47792633547664623918598299666, −6.35413710902425330001465723755, −5.03864915657837585618048477899, −2.96675602736984127460157644008, 3.83538579096647599912719981865, 4.93785631535670169891209178417, 5.87162159242412698248818548088, 8.331695407974657162829287151151, 9.498651377320540499453261903596, 10.82293300995852641350550235410, 12.36580195854198475157715701813, 12.77043670922727425119801816799, 14.41579157887449807636719255788, 15.07775049480717081918669936708

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