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Properties

Label 2-538-269.187-c2-0-27
Degree 22
Conductor 538538
Sign 0.674+0.738i0.674 + 0.738i
Analytic cond. 14.659414.6594
Root an. cond. 3.828763.82876
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (3.13 + 3.13i)3-s + 2i·4-s − 3.10·5-s − 6.26i·6-s + (1.53 − 1.53i)7-s + (2 − 2i)8-s + 10.6i·9-s + (3.10 + 3.10i)10-s − 18.9i·11-s + (−6.26 + 6.26i)12-s − 14.4i·13-s − 3.06·14-s + (−9.73 − 9.73i)15-s − 4·16-s + (0.930 + 0.930i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (1.04 + 1.04i)3-s + 0.5i·4-s − 0.621·5-s − 1.04i·6-s + (0.218 − 0.218i)7-s + (0.250 − 0.250i)8-s + 1.18i·9-s + (0.310 + 0.310i)10-s − 1.72i·11-s + (−0.522 + 0.522i)12-s − 1.10i·13-s − 0.218·14-s + (−0.649 − 0.649i)15-s − 0.250·16-s + (0.0547 + 0.0547i)17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((0.674+0.738i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 + 0.738i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1) L(s)=((0.674+0.738i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: 0.674+0.738i0.674 + 0.738i
Analytic conductor: 14.659414.6594
Root analytic conductor: 3.828763.82876
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ538(187,⋅)\chi_{538} (187, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 538, ( :1), 0.674+0.738i)(2,\ 538,\ (\ :1),\ 0.674 + 0.738i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.6759075871.675907587
L(12)L(\frac12) ≈\approx 1.6759075871.675907587
L(2)L(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+(1+i)T 1 + (1 + i)T
269 1+(−97.9+250.i)T 1 + (-97.9 + 250. i)T
good3 1+(−3.13−3.13i)T+9iT2 1 + (-3.13 - 3.13i)T + 9iT^{2}
5 1+3.10T+25T2 1 + 3.10T + 25T^{2}
7 1+(−1.53+1.53i)T−49iT2 1 + (-1.53 + 1.53i)T - 49iT^{2}
11 1+18.9iT−121T2 1 + 18.9iT - 121T^{2}
13 1+14.4iT−169T2 1 + 14.4iT - 169T^{2}
17 1+(−0.930−0.930i)T+289iT2 1 + (-0.930 - 0.930i)T + 289iT^{2}
19 1+(−12.0+12.0i)T−361iT2 1 + (-12.0 + 12.0i)T - 361iT^{2}
23 1−33.5T+529T2 1 - 33.5T + 529T^{2}
29 1+(14.4−14.4i)T−841iT2 1 + (14.4 - 14.4i)T - 841iT^{2}
31 1+(−27.4+27.4i)T−961iT2 1 + (-27.4 + 27.4i)T - 961iT^{2}
37 1+51.5T+1.36e3T2 1 + 51.5T + 1.36e3T^{2}
41 1−18.4T+1.68e3T2 1 - 18.4T + 1.68e3T^{2}
43 1−41.3iT−1.84e3T2 1 - 41.3iT - 1.84e3T^{2}
47 1−73.3T+2.20e3T2 1 - 73.3T + 2.20e3T^{2}
53 1−32.0T+2.80e3T2 1 - 32.0T + 2.80e3T^{2}
59 1+(−32.0+32.0i)T−3.48e3iT2 1 + (-32.0 + 32.0i)T - 3.48e3iT^{2}
61 1+35.3T+3.72e3T2 1 + 35.3T + 3.72e3T^{2}
67 1+76.5T+4.48e3T2 1 + 76.5T + 4.48e3T^{2}
71 1+(9.32−9.32i)T−5.04e3iT2 1 + (9.32 - 9.32i)T - 5.04e3iT^{2}
73 1−52.8iT−5.32e3T2 1 - 52.8iT - 5.32e3T^{2}
79 1+79.4iT−6.24e3T2 1 + 79.4iT - 6.24e3T^{2}
83 1+(−77.1+77.1i)T−6.88e3iT2 1 + (-77.1 + 77.1i)T - 6.88e3iT^{2}
89 1+84.5iT−7.92e3T2 1 + 84.5iT - 7.92e3T^{2}
97 1+29.7iT−9.40e3T2 1 + 29.7iT - 9.40e3T^{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

−10.58862907509601506813165500710, −9.501156852358923944567198014088, −8.789586318616803142607108154000, −8.174428663915889170912851658163, −7.37525162627273400020908620230, −5.65964398049391510491112171782, −4.41349766319451109308314963210, −3.34128030802501250788312474639, −2.92506482783089798914960716548, −0.74249137586371558283404848791, 1.43081925862586820565266088153, 2.38242473075919721398577249538, 3.94858918170318408684334990426, 5.18261390705458828358292872019, 6.80971820369316564903122206305, 7.21417806971195392843615937314, 7.908047241743502993561719577368, 8.837136357494936680149841786433, 9.441759407641860254229333990275, 10.54711174565177684632263535576

Graph of the ZZ-function along the critical line