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Properties

Label 2-538-269.82-c2-0-17
Degree 22
Conductor 538538
Sign 0.531−0.847i0.531 - 0.847i
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 + (−2.38 + 2.38i)3-s − 2i·4-s − 4.67·5-s − 4.77i·6-s + (4.02 + 4.02i)7-s + (2 + 2i)8-s − 2.41i·9-s + (4.67 − 4.67i)10-s − 0.758i·11-s + (4.77 + 4.77i)12-s − 22.3i·13-s − 8.04·14-s + (11.1 − 11.1i)15-s − 4·16-s + (7.71 − 7.71i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.796 + 0.796i)3-s − 0.5i·4-s − 0.934·5-s − 0.796i·6-s + (0.574 + 0.574i)7-s + (0.250 + 0.250i)8-s − 0.268i·9-s + (0.467 − 0.467i)10-s − 0.0689i·11-s + (0.398 + 0.398i)12-s − 1.71i·13-s − 0.574·14-s + (0.744 − 0.744i)15-s − 0.250·16-s + (0.453 − 0.453i)17-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=((0.531−0.847i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+1) L(s)=((0.531−0.847i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: 0.531−0.847i0.531 - 0.847i
Analytic conductor: 14.659414.6594
Root analytic conductor: 3.828763.82876
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ538(351,⋅)\chi_{538} (351, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 538, ( :1), 0.531−0.847i)(2,\ 538,\ (\ :1),\ 0.531 - 0.847i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.78241680240.7824168024
L(12)L(\frac12) ≈\approx 0.78241680240.7824168024
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+(−141.−228.i)T 1 + (-141. - 228. i)T
good3 1+(2.38−2.38i)T−9iT2 1 + (2.38 - 2.38i)T - 9iT^{2}
5 1+4.67T+25T2 1 + 4.67T + 25T^{2}
7 1+(−4.02−4.02i)T+49iT2 1 + (-4.02 - 4.02i)T + 49iT^{2}
11 1+0.758iT−121T2 1 + 0.758iT - 121T^{2}
13 1+22.3iT−169T2 1 + 22.3iT - 169T^{2}
17 1+(−7.71+7.71i)T−289iT2 1 + (-7.71 + 7.71i)T - 289iT^{2}
19 1+(10.4+10.4i)T+361iT2 1 + (10.4 + 10.4i)T + 361iT^{2}
23 1−25.6T+529T2 1 - 25.6T + 529T^{2}
29 1+(−9.53−9.53i)T+841iT2 1 + (-9.53 - 9.53i)T + 841iT^{2}
31 1+(−11.6−11.6i)T+961iT2 1 + (-11.6 - 11.6i)T + 961iT^{2}
37 1+4.88T+1.36e3T2 1 + 4.88T + 1.36e3T^{2}
41 1−66.2T+1.68e3T2 1 - 66.2T + 1.68e3T^{2}
43 1−57.9iT−1.84e3T2 1 - 57.9iT - 1.84e3T^{2}
47 1−47.1T+2.20e3T2 1 - 47.1T + 2.20e3T^{2}
53 1+45.0T+2.80e3T2 1 + 45.0T + 2.80e3T^{2}
59 1+(−80.1−80.1i)T+3.48e3iT2 1 + (-80.1 - 80.1i)T + 3.48e3iT^{2}
61 1+109.T+3.72e3T2 1 + 109.T + 3.72e3T^{2}
67 1−52.7T+4.48e3T2 1 - 52.7T + 4.48e3T^{2}
71 1+(10.6+10.6i)T+5.04e3iT2 1 + (10.6 + 10.6i)T + 5.04e3iT^{2}
73 1−52.6iT−5.32e3T2 1 - 52.6iT - 5.32e3T^{2}
79 1−19.2iT−6.24e3T2 1 - 19.2iT - 6.24e3T^{2}
83 1+(26.6+26.6i)T+6.88e3iT2 1 + (26.6 + 26.6i)T + 6.88e3iT^{2}
89 1+166.iT−7.92e3T2 1 + 166. iT - 7.92e3T^{2}
97 1−176.iT−9.40e3T2 1 - 176. iT - 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

−10.81314719771237677354164412323, −9.991317034853016243240363691535, −8.892137865007743069772345236615, −8.052163742106099773466751802802, −7.36889646865755654535389676773, −5.95767790992550883330745804754, −5.20990920249904817327887803612, −4.44306139824753528687618410364, −2.91345502274632273150115574531, −0.67139004198237037588432616322, 0.75988692116627327658987536307, 1.89972569704000230973457452940, 3.75786436272804837069488695501, 4.55359739280939927539311000257, 6.10543531730672660242850312132, 7.07444596509136725792685521478, 7.65686344049280742104046899609, 8.629990768648368788454601454600, 9.634761525414367076926627005061, 10.92737903556496225607333932359

Graph of the ZZ-function along the critical line