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Properties

Label 2-538-269.100-c1-0-1
Degree 22
Conductor 538538
Sign −0.934−0.356i-0.934 - 0.356i
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 + (−2.06 − 0.0483i)3-s + (−0.930 − 0.366i)4-s + (−0.765 − 2.90i)5-s + (0.431 − 2.01i)6-s + (0.890 − 1.98i)7-s + (0.533 − 0.845i)8-s + (1.24 + 0.0584i)9-s + (2.99 − 0.210i)10-s + (−0.647 + 0.727i)11-s + (1.89 + 0.799i)12-s + (0.223 + 3.17i)13-s + (1.78 + 1.24i)14-s + (1.43 + 6.01i)15-s + (0.731 + 0.681i)16-s + (−4.38 + 3.89i)17-s + ⋯
L(s)  = 1  + (−0.131 + 0.694i)2-s + (−1.18 − 0.0278i)3-s + (−0.465 − 0.183i)4-s + (−0.342 − 1.29i)5-s + (0.176 − 0.822i)6-s + (0.336 − 0.749i)7-s + (0.188 − 0.299i)8-s + (0.415 + 0.0194i)9-s + (0.947 − 0.0667i)10-s + (−0.195 + 0.219i)11-s + (0.548 + 0.230i)12-s + (0.0620 + 0.881i)13-s + (0.476 + 0.332i)14-s + (0.371 + 1.55i)15-s + (0.182 + 0.170i)16-s + (−1.06 + 0.945i)17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((−0.934−0.356i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1/2) L(s)=((−0.934−0.356i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: −0.934−0.356i-0.934 - 0.356i
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), −0.934−0.356i)(2,\ 538,\ (\ :1/2),\ -0.934 - 0.356i)

Particular Values

L(1)L(1) ≈\approx 0.0249287+0.135341i0.0249287 + 0.135341i
L(12)L(\frac12) ≈\approx 0.0249287+0.135341i0.0249287 + 0.135341i
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+(15.4+5.60i)T 1 + (15.4 + 5.60i)T
good3 1+(2.06+0.0483i)T+(2.99+0.140i)T2 1 + (2.06 + 0.0483i)T + (2.99 + 0.140i)T^{2}
5 1+(0.765+2.90i)T+(−4.34+2.46i)T2 1 + (0.765 + 2.90i)T + (-4.34 + 2.46i)T^{2}
7 1+(−0.890+1.98i)T+(−4.65−5.23i)T2 1 + (-0.890 + 1.98i)T + (-4.65 - 5.23i)T^{2}
11 1+(0.647−0.727i)T+(−1.28−10.9i)T2 1 + (0.647 - 0.727i)T + (-1.28 - 10.9i)T^{2}
13 1+(−0.223−3.17i)T+(−12.8+1.82i)T2 1 + (-0.223 - 3.17i)T + (-12.8 + 1.82i)T^{2}
17 1+(4.38−3.89i)T+(1.98−16.8i)T2 1 + (4.38 - 3.89i)T + (1.98 - 16.8i)T^{2}
19 1+(1.29−0.857i)T+(7.37−17.5i)T2 1 + (1.29 - 0.857i)T + (7.37 - 17.5i)T^{2}
23 1+(−0.0751+3.20i)T+(−22.9−1.07i)T2 1 + (-0.0751 + 3.20i)T + (-22.9 - 1.07i)T^{2}
29 1+(2.61−4.60i)T+(−14.8−24.8i)T2 1 + (2.61 - 4.60i)T + (-14.8 - 24.8i)T^{2}
31 1+(2.84−0.335i)T+(30.1−7.20i)T2 1 + (2.84 - 0.335i)T + (30.1 - 7.20i)T^{2}
37 1+(1.28+2.39i)T+(−20.4+30.8i)T2 1 + (1.28 + 2.39i)T + (-20.4 + 30.8i)T^{2}
41 1+(2.32−0.440i)T+(38.1−15.0i)T2 1 + (2.32 - 0.440i)T + (38.1 - 15.0i)T^{2}
43 1+(−4.44−2.52i)T+(22.0+36.8i)T2 1 + (-4.44 - 2.52i)T + (22.0 + 36.8i)T^{2}
47 1+(0.272−0.0788i)T+(39.7−25.0i)T2 1 + (0.272 - 0.0788i)T + (39.7 - 25.0i)T^{2}
53 1+(7.98−10.8i)T+(−15.9−50.5i)T2 1 + (7.98 - 10.8i)T + (-15.9 - 50.5i)T^{2}
59 1+(1.59−4.05i)T+(−43.1−40.2i)T2 1 + (1.59 - 4.05i)T + (-43.1 - 40.2i)T^{2}
61 1+(6.91−8.54i)T+(−12.7−59.6i)T2 1 + (6.91 - 8.54i)T + (-12.7 - 59.6i)T^{2}
67 1+(−2.68+1.05i)T+(49.0−45.6i)T2 1 + (-2.68 + 1.05i)T + (49.0 - 45.6i)T^{2}
71 1+(6.75+9.67i)T+(−24.4+66.6i)T2 1 + (6.75 + 9.67i)T + (-24.4 + 66.6i)T^{2}
73 1+(−0.149−0.406i)T+(−55.6+47.2i)T2 1 + (-0.149 - 0.406i)T + (-55.6 + 47.2i)T^{2}
79 1+(1.94−1.50i)T+(20.1−76.3i)T2 1 + (1.94 - 1.50i)T + (20.1 - 76.3i)T^{2}
83 1+(0.804−8.54i)T+(−81.5−15.4i)T2 1 + (0.804 - 8.54i)T + (-81.5 - 15.4i)T^{2}
89 1+(−0.744+4.49i)T+(−84.2−28.6i)T2 1 + (-0.744 + 4.49i)T + (-84.2 - 28.6i)T^{2}
97 1+(−6.95−8.60i)T+(−20.3+94.8i)T2 1 + (-6.95 - 8.60i)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

−11.04268486670315221085120683192, −10.58467207024498496898738264254, −9.172762296013599672881691532517, −8.581080355594606761400401526056, −7.52988963828064339581071034240, −6.58101763522647673293673202024, −5.70296782862142400267808186694, −4.63285235947857113174186276613, −4.25804613242130860304309800052, −1.39240794649207275132172270612, 0.10192917759816884697659065899, 2.34113347458488596798725849391, 3.36978791787020546121287676388, 4.84695977152715975099902385786, 5.71607233314443666787823721410, 6.67272522681332599868755867190, 7.68866392861177054013052489512, 8.780894744811539387548355578483, 9.911700947055786357085789839542, 10.79766353062664221048410368796

Graph of the ZZ-function along the critical line