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Properties

Label 2-538-269.187-c2-0-9
Degree 22
Conductor 538538
Sign −0.205+0.978i-0.205 + 0.978i
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 + (−4.16 − 4.16i)3-s + 2i·4-s − 6.39·5-s + 8.33i·6-s + (−6.86 + 6.86i)7-s + (2 − 2i)8-s + 25.7i·9-s + (6.39 + 6.39i)10-s − 13.6i·11-s + (8.33 − 8.33i)12-s + 20.6i·13-s + 13.7·14-s + (26.6 + 26.6i)15-s − 4·16-s + (−12.6 − 12.6i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−1.38 − 1.38i)3-s + 0.5i·4-s − 1.27·5-s + 1.38i·6-s + (−0.981 + 0.981i)7-s + (0.250 − 0.250i)8-s + 2.85i·9-s + (0.639 + 0.639i)10-s − 1.23i·11-s + (0.694 − 0.694i)12-s + 1.58i·13-s + 0.981·14-s + (1.77 + 1.77i)15-s − 0.250·16-s + (−0.742 − 0.742i)17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((−0.205+0.978i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1) L(s)=((−0.205+0.978i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: −0.205+0.978i-0.205 + 0.978i
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), −0.205+0.978i)(2,\ 538,\ (\ :1),\ -0.205 + 0.978i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.13778920300.1377892030
L(12)L(\frac12) ≈\approx 0.13778920300.1377892030
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+(−260.+66.7i)T 1 + (-260. + 66.7i)T
good3 1+(4.16+4.16i)T+9iT2 1 + (4.16 + 4.16i)T + 9iT^{2}
5 1+6.39T+25T2 1 + 6.39T + 25T^{2}
7 1+(6.86−6.86i)T−49iT2 1 + (6.86 - 6.86i)T - 49iT^{2}
11 1+13.6iT−121T2 1 + 13.6iT - 121T^{2}
13 1−20.6iT−169T2 1 - 20.6iT - 169T^{2}
17 1+(12.6+12.6i)T+289iT2 1 + (12.6 + 12.6i)T + 289iT^{2}
19 1+(−7.34+7.34i)T−361iT2 1 + (-7.34 + 7.34i)T - 361iT^{2}
23 1+24.6T+529T2 1 + 24.6T + 529T^{2}
29 1+(33.1−33.1i)T−841iT2 1 + (33.1 - 33.1i)T - 841iT^{2}
31 1+(24.0−24.0i)T−961iT2 1 + (24.0 - 24.0i)T - 961iT^{2}
37 1−14.3T+1.36e3T2 1 - 14.3T + 1.36e3T^{2}
41 1+16.7T+1.68e3T2 1 + 16.7T + 1.68e3T^{2}
43 1−59.3iT−1.84e3T2 1 - 59.3iT - 1.84e3T^{2}
47 1+38.1T+2.20e3T2 1 + 38.1T + 2.20e3T^{2}
53 1+80.2T+2.80e3T2 1 + 80.2T + 2.80e3T^{2}
59 1+(−30.6+30.6i)T−3.48e3iT2 1 + (-30.6 + 30.6i)T - 3.48e3iT^{2}
61 1−44.1T+3.72e3T2 1 - 44.1T + 3.72e3T^{2}
67 1+70.1T+4.48e3T2 1 + 70.1T + 4.48e3T^{2}
71 1+(−69.2+69.2i)T−5.04e3iT2 1 + (-69.2 + 69.2i)T - 5.04e3iT^{2}
73 1+23.2iT−5.32e3T2 1 + 23.2iT - 5.32e3T^{2}
79 1+85.9iT−6.24e3T2 1 + 85.9iT - 6.24e3T^{2}
83 1+(−19.2+19.2i)T−6.88e3iT2 1 + (-19.2 + 19.2i)T - 6.88e3iT^{2}
89 1+99.4iT−7.92e3T2 1 + 99.4iT - 7.92e3T^{2}
97 1+40.2iT−9.40e3T2 1 + 40.2iT - 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

−10.95587273704442995935731655943, −9.386221558361746291652771183432, −8.521288224746863164180482025976, −7.54383913668963749022686229576, −6.75412217010387425790887272805, −6.05690845717952754217966944091, −4.79326917445691444604210913837, −3.24121527300247061748122171723, −1.82968379296503967318778428120, −0.22569720990034555116882429828, 0.38024788179736110048736649373, 3.77747512255128337056463563940, 4.07749919839283340367046228778, 5.32474585405366696602306979524, 6.24937463887289109544242995701, 7.20163814617622014050878055931, 8.057982073247081560730908247949, 9.551462344974473820328897136619, 10.01619486933192417969049141811, 10.65741974097457166026194167430

Graph of the ZZ-function along the critical line