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Properties

Label 2-507-39.2-c1-0-34
Degree 22
Conductor 507507
Sign −0.642+0.766i-0.642 + 0.766i
Analytic cond. 4.048414.04841
Root an. cond. 2.012062.01206
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  + (0.619 − 2.31i)2-s + (1.64 − 0.529i)3-s + (−3.23 − 1.86i)4-s + (1.69 + 1.69i)5-s + (−0.202 − 4.14i)6-s + (1.36 − 0.366i)7-s + (−2.93 + 2.93i)8-s + (2.43 − 1.74i)9-s + (4.96 − 2.86i)10-s + (−1.69 − 0.453i)11-s + (−6.31 − 1.36i)12-s − 3.38i·14-s + (3.68 + 1.89i)15-s + (1.23 + 2.13i)16-s + (−1.07 + 1.85i)17-s + (−2.52 − 6.72i)18-s + ⋯
L(s)  = 1  + (0.438 − 1.63i)2-s + (0.952 − 0.305i)3-s + (−1.61 − 0.933i)4-s + (0.757 + 0.757i)5-s + (−0.0826 − 1.69i)6-s + (0.516 − 0.138i)7-s + (−1.03 + 1.03i)8-s + (0.813 − 0.582i)9-s + (1.56 − 0.906i)10-s + (−0.510 − 0.136i)11-s + (−1.82 − 0.394i)12-s − 0.904i·14-s + (0.952 + 0.489i)15-s + (0.308 + 0.533i)16-s + (−0.260 + 0.450i)17-s + (−0.595 − 1.58i)18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((−0.642+0.766i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((−0.642+0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: −0.642+0.766i-0.642 + 0.766i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(80,⋅)\chi_{507} (80, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), −0.642+0.766i)(2,\ 507,\ (\ :1/2),\ -0.642 + 0.766i)

Particular Values

L(1)L(1) ≈\approx 1.08436−2.32476i1.08436 - 2.32476i
L(12)L(\frac12) ≈\approx 1.08436−2.32476i1.08436 - 2.32476i
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)
bad3 1+(−1.64+0.529i)T 1 + (-1.64 + 0.529i)T
13 1 1
good2 1+(−0.619+2.31i)T+(−1.73−i)T2 1 + (-0.619 + 2.31i)T + (-1.73 - i)T^{2}
5 1+(−1.69−1.69i)T+5iT2 1 + (-1.69 - 1.69i)T + 5iT^{2}
7 1+(−1.36+0.366i)T+(6.06−3.5i)T2 1 + (-1.36 + 0.366i)T + (6.06 - 3.5i)T^{2}
11 1+(1.69+0.453i)T+(9.52+5.5i)T2 1 + (1.69 + 0.453i)T + (9.52 + 5.5i)T^{2}
17 1+(1.07−1.85i)T+(−8.5−14.7i)T2 1 + (1.07 - 1.85i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.267−i)T+(−16.4+9.5i)T2 1 + (-0.267 - i)T + (-16.4 + 9.5i)T^{2}
23 1+(−11.5+19.9i)T2 1 + (-11.5 + 19.9i)T^{2}
29 1+(4.79−2.76i)T+(14.5−25.1i)T2 1 + (4.79 - 2.76i)T + (14.5 - 25.1i)T^{2}
31 1+(4.46−4.46i)T−31iT2 1 + (4.46 - 4.46i)T - 31iT^{2}
37 1+(−1.76+6.59i)T+(−32.0−18.5i)T2 1 + (-1.76 + 6.59i)T + (-32.0 - 18.5i)T^{2}
41 1+(0.166−0.619i)T+(−35.5−20.5i)T2 1 + (0.166 - 0.619i)T + (-35.5 - 20.5i)T^{2}
43 1+(−7.09−4.09i)T+(21.5+37.2i)T2 1 + (-7.09 - 4.09i)T + (21.5 + 37.2i)T^{2}
47 1+(6.77−6.77i)T−47iT2 1 + (6.77 - 6.77i)T - 47iT^{2}
53 1−4.62iT−53T2 1 - 4.62iT - 53T^{2}
59 1+(−1.23−4.62i)T+(−51.0+29.5i)T2 1 + (-1.23 - 4.62i)T + (-51.0 + 29.5i)T^{2}
61 1+(−3.5+6.06i)T+(−30.5−52.8i)T2 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2}
67 1+(−8.46−2.26i)T+(58.0+33.5i)T2 1 + (-8.46 - 2.26i)T + (58.0 + 33.5i)T^{2}
71 1+(−4.62+1.23i)T+(61.4−35.5i)T2 1 + (-4.62 + 1.23i)T + (61.4 - 35.5i)T^{2}
73 1+(−6.09−6.09i)T+73iT2 1 + (-6.09 - 6.09i)T + 73iT^{2}
79 1−2T+79T2 1 - 2T + 79T^{2}
83 1+(1.23+1.23i)T+83iT2 1 + (1.23 + 1.23i)T + 83iT^{2}
89 1+(9.70+2.60i)T+(77.0+44.5i)T2 1 + (9.70 + 2.60i)T + (77.0 + 44.5i)T^{2}
97 1+(3.36+12.5i)T+(−84.0+48.5i)T2 1 + (3.36 + 12.5i)T + (-84.0 + 48.5i)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

−10.76711712578824688547979554424, −9.868013591030098162484528497900, −9.219086687085310586451207518819, −8.107392332221407185430835025230, −7.00583069313306420897137090786, −5.66836629336547823930856055010, −4.36308546912651289176207392987, −3.31833748058406673864514898220, −2.42736156810364704307919461499, −1.55517106438764427463189720281, 2.11921062660247027095508527948, 3.87090041505874895280465307412, 4.97277409753283044878408069065, 5.42068457535407245861796797120, 6.72883844000573286779618515790, 7.71392367140448139630699517104, 8.329629940188430756196260887069, 9.174409080768675891859802830699, 9.783031085160026904467692600584, 11.17541018111125974572322737523

Graph of the ZZ-function along the critical line