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Properties

Label 2-57-57.2-c1-0-3
Degree 22
Conductor 5757
Sign 0.0291+0.999i0.0291 + 0.999i
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  + (0.448 − 2.54i)2-s + (0.504 + 1.65i)3-s + (−4.37 − 1.59i)4-s + (−0.533 − 1.46i)5-s + (4.43 − 0.538i)6-s + (1.49 + 2.59i)7-s + (−3.42 + 5.93i)8-s + (−2.49 + 1.67i)9-s + (−3.96 + 0.699i)10-s + (1.05 + 0.611i)11-s + (0.432 − 8.05i)12-s + (0.203 + 0.242i)13-s + (7.25 − 2.64i)14-s + (2.16 − 1.62i)15-s + (6.41 + 5.38i)16-s + (−5.00 − 0.882i)17-s + ⋯
L(s)  = 1  + (0.316 − 1.79i)2-s + (0.291 + 0.956i)3-s + (−2.18 − 0.796i)4-s + (−0.238 − 0.655i)5-s + (1.81 − 0.219i)6-s + (0.565 + 0.979i)7-s + (−1.21 + 2.09i)8-s + (−0.830 + 0.556i)9-s + (−1.25 + 0.221i)10-s + (0.319 + 0.184i)11-s + (0.124 − 2.32i)12-s + (0.0564 + 0.0672i)13-s + (1.93 − 0.706i)14-s + (0.558 − 0.419i)15-s + (1.60 + 1.34i)16-s + (−1.21 − 0.214i)17-s + ⋯

Functional equation

Λ(s)=(57s/2 ΓC(s) L(s)=((0.0291+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0291 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(57s/2 ΓC(s+1/2) L(s)=((0.0291+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0291 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5757    =    3⋅193 \cdot 19
Sign: 0.0291+0.999i0.0291 + 0.999i
Analytic conductor: 0.4551470.455147
Root analytic conductor: 0.6746460.674646
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ57(2,⋅)\chi_{57} (2, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 57, ( :1/2), 0.0291+0.999i)(2,\ 57,\ (\ :1/2),\ 0.0291 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.680820−0.661248i0.680820 - 0.661248i
L(12)L(\frac12) ≈\approx 0.680820−0.661248i0.680820 - 0.661248i
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+(−0.504−1.65i)T 1 + (-0.504 - 1.65i)T
19 1+(0.461+4.33i)T 1 + (0.461 + 4.33i)T
good2 1+(−0.448+2.54i)T+(−1.87−0.684i)T2 1 + (-0.448 + 2.54i)T + (-1.87 - 0.684i)T^{2}
5 1+(0.533+1.46i)T+(−3.83+3.21i)T2 1 + (0.533 + 1.46i)T + (-3.83 + 3.21i)T^{2}
7 1+(−1.49−2.59i)T+(−3.5+6.06i)T2 1 + (-1.49 - 2.59i)T + (-3.5 + 6.06i)T^{2}
11 1+(−1.05−0.611i)T+(5.5+9.52i)T2 1 + (-1.05 - 0.611i)T + (5.5 + 9.52i)T^{2}
13 1+(−0.203−0.242i)T+(−2.25+12.8i)T2 1 + (-0.203 - 0.242i)T + (-2.25 + 12.8i)T^{2}
17 1+(5.00+0.882i)T+(15.9+5.81i)T2 1 + (5.00 + 0.882i)T + (15.9 + 5.81i)T^{2}
23 1+(−0.211+0.581i)T+(−17.6−14.7i)T2 1 + (-0.211 + 0.581i)T + (-17.6 - 14.7i)T^{2}
29 1+(−0.204−1.16i)T+(−27.2+9.91i)T2 1 + (-0.204 - 1.16i)T + (-27.2 + 9.91i)T^{2}
31 1+(4.50−2.59i)T+(15.5−26.8i)T2 1 + (4.50 - 2.59i)T + (15.5 - 26.8i)T^{2}
37 1−3.64iT−37T2 1 - 3.64iT - 37T^{2}
41 1+(−0.118−0.0991i)T+(7.11+40.3i)T2 1 + (-0.118 - 0.0991i)T + (7.11 + 40.3i)T^{2}
43 1+(−8.99+3.27i)T+(32.9−27.6i)T2 1 + (-8.99 + 3.27i)T + (32.9 - 27.6i)T^{2}
47 1+(−8.03+1.41i)T+(44.1−16.0i)T2 1 + (-8.03 + 1.41i)T + (44.1 - 16.0i)T^{2}
53 1+(−2.88−1.04i)T+(40.6+34.0i)T2 1 + (-2.88 - 1.04i)T + (40.6 + 34.0i)T^{2}
59 1+(0.468−2.65i)T+(−55.4−20.1i)T2 1 + (0.468 - 2.65i)T + (-55.4 - 20.1i)T^{2}
61 1+(6.47+2.35i)T+(46.7+39.2i)T2 1 + (6.47 + 2.35i)T + (46.7 + 39.2i)T^{2}
67 1+(−8.96+1.58i)T+(62.9−22.9i)T2 1 + (-8.96 + 1.58i)T + (62.9 - 22.9i)T^{2}
71 1+(12.9−4.71i)T+(54.3−45.6i)T2 1 + (12.9 - 4.71i)T + (54.3 - 45.6i)T^{2}
73 1+(0.335+0.281i)T+(12.6+71.8i)T2 1 + (0.335 + 0.281i)T + (12.6 + 71.8i)T^{2}
79 1+(0.940−1.12i)T+(−13.7−77.7i)T2 1 + (0.940 - 1.12i)T + (-13.7 - 77.7i)T^{2}
83 1+(−9.46+5.46i)T+(41.5−71.8i)T2 1 + (-9.46 + 5.46i)T + (41.5 - 71.8i)T^{2}
89 1+(6.02−5.05i)T+(15.4−87.6i)T2 1 + (6.02 - 5.05i)T + (15.4 - 87.6i)T^{2}
97 1+(18.2+3.22i)T+(91.1+33.1i)T2 1 + (18.2 + 3.22i)T + (91.1 + 33.1i)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

−14.74730821244428315079917712677, −13.61043964363685890483144427215, −12.41141440792709904896417896154, −11.48121859348336131983254137109, −10.63069376270207414385349422280, −9.103809746490045656224312906612, −8.795429182988543433929697451308, −5.15939751814214065412083682445, −4.25533746471329045740840367005, −2.48419763154511860551357595394, 4.03065015756233248617212531846, 5.98013733297923467174623208320, 7.08965394851649268025015528767, 7.75258961857716889152217272859, 8.935396267977817078411835917273, 11.03259986898804825973168957545, 12.74528398877480574271040846241, 13.79365965909385198508441716796, 14.40395766608295190422883528512, 15.24598787769030660608086717296

Graph of the ZZ-function along the critical line