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Properties

Label 2-57-57.32-c1-0-2
Degree 22
Conductor 5757
Sign 0.997−0.0646i0.997 - 0.0646i
Analytic cond. 0.4551470.455147
Root an. cond. 0.6746460.674646
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.745 + 0.625i)2-s + (0.0227 − 1.73i)3-s + (−0.182 + 1.03i)4-s + (3.79 − 0.668i)5-s + (1.06 + 1.30i)6-s + (−0.469 + 0.813i)7-s + (−1.48 − 2.57i)8-s + (−2.99 − 0.0789i)9-s + (−2.40 + 2.87i)10-s + (−3.44 + 1.99i)11-s + (1.79 + 0.340i)12-s + (−0.353 − 0.970i)13-s + (−0.158 − 0.901i)14-s + (−1.07 − 6.58i)15-s + (0.740 + 0.269i)16-s + (0.804 + 0.958i)17-s + ⋯
L(s)  = 1  + (−0.527 + 0.442i)2-s + (0.0131 − 0.999i)3-s + (−0.0913 + 0.518i)4-s + (1.69 − 0.298i)5-s + (0.435 + 0.533i)6-s + (−0.177 + 0.307i)7-s + (−0.525 − 0.909i)8-s + (−0.999 − 0.0263i)9-s + (−0.761 + 0.907i)10-s + (−1.03 + 0.600i)11-s + (0.516 + 0.0981i)12-s + (−0.0979 − 0.269i)13-s + (−0.0424 − 0.240i)14-s + (−0.276 − 1.69i)15-s + (0.185 + 0.0673i)16-s + (0.195 + 0.232i)17-s + ⋯

Functional equation

Λ(s)=(57s/2 ΓC(s) L(s)=((0.997−0.0646i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0646i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(57s/2 ΓC(s+1/2) L(s)=((0.997−0.0646i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0646i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5757    =    3⋅193 \cdot 19
Sign: 0.997−0.0646i0.997 - 0.0646i
Analytic conductor: 0.4551470.455147
Root analytic conductor: 0.6746460.674646
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ57(32,⋅)\chi_{57} (32, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 57, ( :1/2), 0.997−0.0646i)(2,\ 57,\ (\ :1/2),\ 0.997 - 0.0646i)

Particular Values

L(1)L(1) ≈\approx 0.752142+0.0243217i0.752142 + 0.0243217i
L(12)L(\frac12) ≈\approx 0.752142+0.0243217i0.752142 + 0.0243217i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(−0.0227+1.73i)T 1 + (-0.0227 + 1.73i)T
19 1+(4.22−1.09i)T 1 + (4.22 - 1.09i)T
good2 1+(0.745−0.625i)T+(0.347−1.96i)T2 1 + (0.745 - 0.625i)T + (0.347 - 1.96i)T^{2}
5 1+(−3.79+0.668i)T+(4.69−1.71i)T2 1 + (-3.79 + 0.668i)T + (4.69 - 1.71i)T^{2}
7 1+(0.469−0.813i)T+(−3.5−6.06i)T2 1 + (0.469 - 0.813i)T + (-3.5 - 6.06i)T^{2}
11 1+(3.44−1.99i)T+(5.5−9.52i)T2 1 + (3.44 - 1.99i)T + (5.5 - 9.52i)T^{2}
13 1+(0.353+0.970i)T+(−9.95+8.35i)T2 1 + (0.353 + 0.970i)T + (-9.95 + 8.35i)T^{2}
17 1+(−0.804−0.958i)T+(−2.95+16.7i)T2 1 + (-0.804 - 0.958i)T + (-2.95 + 16.7i)T^{2}
23 1+(−0.477−0.0842i)T+(21.6+7.86i)T2 1 + (-0.477 - 0.0842i)T + (21.6 + 7.86i)T^{2}
29 1+(3.32+2.79i)T+(5.03+28.5i)T2 1 + (3.32 + 2.79i)T + (5.03 + 28.5i)T^{2}
31 1+(−4.26−2.46i)T+(15.5+26.8i)T2 1 + (-4.26 - 2.46i)T + (15.5 + 26.8i)T^{2}
37 1+1.10iT−37T2 1 + 1.10iT - 37T^{2}
41 1+(2.04+0.745i)T+(31.4+26.3i)T2 1 + (2.04 + 0.745i)T + (31.4 + 26.3i)T^{2}
43 1+(1.00+5.68i)T+(−40.4+14.7i)T2 1 + (1.00 + 5.68i)T + (-40.4 + 14.7i)T^{2}
47 1+(−4.96+5.91i)T+(−8.16−46.2i)T2 1 + (-4.96 + 5.91i)T + (-8.16 - 46.2i)T^{2}
53 1+(1.34−7.60i)T+(−49.8−18.1i)T2 1 + (1.34 - 7.60i)T + (-49.8 - 18.1i)T^{2}
59 1+(−5.26+4.41i)T+(10.2−58.1i)T2 1 + (-5.26 + 4.41i)T + (10.2 - 58.1i)T^{2}
61 1+(−0.740+4.19i)T+(−57.3−20.8i)T2 1 + (-0.740 + 4.19i)T + (-57.3 - 20.8i)T^{2}
67 1+(−0.918+1.09i)T+(−11.6−65.9i)T2 1 + (-0.918 + 1.09i)T + (-11.6 - 65.9i)T^{2}
71 1+(0.438+2.48i)T+(−66.7+24.2i)T2 1 + (0.438 + 2.48i)T + (-66.7 + 24.2i)T^{2}
73 1+(−8.28−3.01i)T+(55.9+46.9i)T2 1 + (-8.28 - 3.01i)T + (55.9 + 46.9i)T^{2}
79 1+(1.50−4.12i)T+(−60.5−50.7i)T2 1 + (1.50 - 4.12i)T + (-60.5 - 50.7i)T^{2}
83 1+(−3.66−2.11i)T+(41.5+71.8i)T2 1 + (-3.66 - 2.11i)T + (41.5 + 71.8i)T^{2}
89 1+(7.56−2.75i)T+(68.1−57.2i)T2 1 + (7.56 - 2.75i)T + (68.1 - 57.2i)T^{2}
97 1+(6.91+8.24i)T+(−16.8+95.5i)T2 1 + (6.91 + 8.24i)T + (-16.8 + 95.5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−15.33815943906876514476904918940, −13.84153608934125669909046272436, −12.95602643933962346507748373240, −12.39446016659406824477569125268, −10.31253276706278026752767350073, −9.133155658137541700741982231298, −8.059536609143553624012683588058, −6.71241508658591204233581585497, −5.57481108667172655730027824333, −2.38270525398588913453961998960, 2.56207921107022638495579854103, 5.15037944379188205793096949226, 6.16061506480817802602742977908, 8.688332631782797584892628433426, 9.730233301008090417395339116466, 10.34729716360187648883185241093, 11.15275347054492846082042707525, 13.28972213459351100389050163580, 14.16347240811914020412538276146, 15.09231346072462553354643257150

Graph of the ZZ-function along the critical line