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Properties

Label 2-538-269.100-c1-0-6
Degree 22
Conductor 538538
Sign −0.538−0.842i-0.538 - 0.842i
Analytic cond. 4.295954.29595
Root an. cond. 2.072662.07266
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  + (−0.186 + 0.982i)2-s + (−3.09 − 0.0725i)3-s + (−0.930 − 0.366i)4-s + (0.532 + 2.01i)5-s + (0.648 − 3.02i)6-s + (0.698 − 1.55i)7-s + (0.533 − 0.845i)8-s + (6.58 + 0.308i)9-s + (−2.08 + 0.146i)10-s + (1.67 − 1.88i)11-s + (2.85 + 1.20i)12-s + (0.229 + 3.25i)13-s + (1.39 + 0.976i)14-s + (−1.50 − 6.28i)15-s + (0.731 + 0.681i)16-s + (5.53 − 4.92i)17-s + ⋯
L(s)  = 1  + (−0.131 + 0.694i)2-s + (−1.78 − 0.0419i)3-s + (−0.465 − 0.183i)4-s + (0.238 + 0.902i)5-s + (0.264 − 1.23i)6-s + (0.264 − 0.588i)7-s + (0.188 − 0.299i)8-s + (2.19 + 0.102i)9-s + (−0.658 + 0.0463i)10-s + (0.504 − 0.567i)11-s + (0.823 + 0.346i)12-s + (0.0636 + 0.903i)13-s + (0.373 + 0.261i)14-s + (−0.387 − 1.62i)15-s + (0.182 + 0.170i)16-s + (1.34 − 1.19i)17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((−0.538−0.842i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1/2) L(s)=((−0.538−0.842i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: −0.538−0.842i-0.538 - 0.842i
Analytic conductor: 4.295954.29595
Root analytic conductor: 2.072662.07266
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ538(369,⋅)\chi_{538} (369, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 538, ( :1/2), −0.538−0.842i)(2,\ 538,\ (\ :1/2),\ -0.538 - 0.842i)

Particular Values

L(1)L(1) ≈\approx 0.318383+0.581518i0.318383 + 0.581518i
L(12)L(\frac12) ≈\approx 0.318383+0.581518i0.318383 + 0.581518i
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)
bad2 1+(0.186−0.982i)T 1 + (0.186 - 0.982i)T
269 1+(9.05+13.6i)T 1 + (9.05 + 13.6i)T
good3 1+(3.09+0.0725i)T+(2.99+0.140i)T2 1 + (3.09 + 0.0725i)T + (2.99 + 0.140i)T^{2}
5 1+(−0.532−2.01i)T+(−4.34+2.46i)T2 1 + (-0.532 - 2.01i)T + (-4.34 + 2.46i)T^{2}
7 1+(−0.698+1.55i)T+(−4.65−5.23i)T2 1 + (-0.698 + 1.55i)T + (-4.65 - 5.23i)T^{2}
11 1+(−1.67+1.88i)T+(−1.28−10.9i)T2 1 + (-1.67 + 1.88i)T + (-1.28 - 10.9i)T^{2}
13 1+(−0.229−3.25i)T+(−12.8+1.82i)T2 1 + (-0.229 - 3.25i)T + (-12.8 + 1.82i)T^{2}
17 1+(−5.53+4.92i)T+(1.98−16.8i)T2 1 + (-5.53 + 4.92i)T + (1.98 - 16.8i)T^{2}
19 1+(7.07−4.69i)T+(7.37−17.5i)T2 1 + (7.07 - 4.69i)T + (7.37 - 17.5i)T^{2}
23 1+(0.0778−3.32i)T+(−22.9−1.07i)T2 1 + (0.0778 - 3.32i)T + (-22.9 - 1.07i)T^{2}
29 1+(4.42−7.80i)T+(−14.8−24.8i)T2 1 + (4.42 - 7.80i)T + (-14.8 - 24.8i)T^{2}
31 1+(−4.01+0.472i)T+(30.1−7.20i)T2 1 + (-4.01 + 0.472i)T + (30.1 - 7.20i)T^{2}
37 1+(−3.75−6.99i)T+(−20.4+30.8i)T2 1 + (-3.75 - 6.99i)T + (-20.4 + 30.8i)T^{2}
41 1+(11.6−2.21i)T+(38.1−15.0i)T2 1 + (11.6 - 2.21i)T + (38.1 - 15.0i)T^{2}
43 1+(0.640+0.363i)T+(22.0+36.8i)T2 1 + (0.640 + 0.363i)T + (22.0 + 36.8i)T^{2}
47 1+(3.40−0.985i)T+(39.7−25.0i)T2 1 + (3.40 - 0.985i)T + (39.7 - 25.0i)T^{2}
53 1+(−4.31+5.88i)T+(−15.9−50.5i)T2 1 + (-4.31 + 5.88i)T + (-15.9 - 50.5i)T^{2}
59 1+(2.33−5.93i)T+(−43.1−40.2i)T2 1 + (2.33 - 5.93i)T + (-43.1 - 40.2i)T^{2}
61 1+(−2.10+2.59i)T+(−12.7−59.6i)T2 1 + (-2.10 + 2.59i)T + (-12.7 - 59.6i)T^{2}
67 1+(−9.96+3.92i)T+(49.0−45.6i)T2 1 + (-9.96 + 3.92i)T + (49.0 - 45.6i)T^{2}
71 1+(−6.58−9.43i)T+(−24.4+66.6i)T2 1 + (-6.58 - 9.43i)T + (-24.4 + 66.6i)T^{2}
73 1+(−2.41−6.57i)T+(−55.6+47.2i)T2 1 + (-2.41 - 6.57i)T + (-55.6 + 47.2i)T^{2}
79 1+(−0.584+0.450i)T+(20.1−76.3i)T2 1 + (-0.584 + 0.450i)T + (20.1 - 76.3i)T^{2}
83 1+(0.441−4.69i)T+(−81.5−15.4i)T2 1 + (0.441 - 4.69i)T + (-81.5 - 15.4i)T^{2}
89 1+(2.74−16.5i)T+(−84.2−28.6i)T2 1 + (2.74 - 16.5i)T + (-84.2 - 28.6i)T^{2}
97 1+(4.18+5.17i)T+(−20.3+94.8i)T2 1 + (4.18 + 5.17i)T + (-20.3 + 94.8i)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

−11.13340213234494690777130985610, −10.30591491559653025593026800306, −9.704903090921604881623964995006, −8.208205100074676997436963569099, −6.96187769513207615974903837591, −6.68503443825819349688587148018, −5.76760130399089200177141899281, −4.88297451721081420499018310513, −3.72852262203814387711427518556, −1.28221478396287476739429848637, 0.59445899620366981316766909742, 1.91532937770597666355720437359, 4.06423617923112397177386521179, 4.94572460359600685392500770451, 5.67592434762625255259832317902, 6.53429848817585709333371739080, 7.988346867721258449002781351328, 8.938597117019317595657123903446, 10.02472495408021199864458993685, 10.56881154880690972125204915066

Graph of the ZZ-function along the critical line