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Properties

Label 2-507-13.9-c1-0-24
Degree 22
Conductor 507507
Sign −0.668−0.743i-0.668 - 0.743i
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  + (1.17 − 2.04i)2-s + (0.5 − 0.866i)3-s + (−1.77 − 3.07i)4-s − 3.69·5-s + (−1.17 − 2.04i)6-s + (−0.400 − 0.694i)7-s − 3.66·8-s + (−0.499 − 0.866i)9-s + (−4.35 + 7.53i)10-s + (−1.42 + 2.46i)11-s − 3.55·12-s − 1.89·14-s + (−1.84 + 3.19i)15-s + (−0.763 + 1.32i)16-s + (−1.46 − 2.54i)17-s − 2.35·18-s + ⋯
L(s)  = 1  + (0.833 − 1.44i)2-s + (0.288 − 0.499i)3-s + (−0.888 − 1.53i)4-s − 1.65·5-s + (−0.481 − 0.833i)6-s + (−0.151 − 0.262i)7-s − 1.29·8-s + (−0.166 − 0.288i)9-s + (−1.37 + 2.38i)10-s + (−0.429 + 0.744i)11-s − 1.02·12-s − 0.505·14-s + (−0.476 + 0.825i)15-s + (−0.190 + 0.330i)16-s + (−0.356 − 0.617i)17-s − 0.555·18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((−0.668−0.743i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((−0.668−0.743i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: −0.668−0.743i-0.668 - 0.743i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(22,⋅)\chi_{507} (22, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), −0.668−0.743i)(2,\ 507,\ (\ :1/2),\ -0.668 - 0.743i)

Particular Values

L(1)L(1) ≈\approx 0.503792+1.13071i0.503792 + 1.13071i
L(12)L(\frac12) ≈\approx 0.503792+1.13071i0.503792 + 1.13071i
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
13 1 1
good2 1+(−1.17+2.04i)T+(−1−1.73i)T2 1 + (-1.17 + 2.04i)T + (-1 - 1.73i)T^{2}
5 1+3.69T+5T2 1 + 3.69T + 5T^{2}
7 1+(0.400+0.694i)T+(−3.5+6.06i)T2 1 + (0.400 + 0.694i)T + (-3.5 + 6.06i)T^{2}
11 1+(1.42−2.46i)T+(−5.5−9.52i)T2 1 + (1.42 - 2.46i)T + (-5.5 - 9.52i)T^{2}
17 1+(1.46+2.54i)T+(−8.5+14.7i)T2 1 + (1.46 + 2.54i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.22+2.11i)T+(−9.5+16.4i)T2 1 + (1.22 + 2.11i)T + (-9.5 + 16.4i)T^{2}
23 1+(−3.89+6.74i)T+(−11.5−19.9i)T2 1 + (-3.89 + 6.74i)T + (-11.5 - 19.9i)T^{2}
29 1+(1.92−3.33i)T+(−14.5−25.1i)T2 1 + (1.92 - 3.33i)T + (-14.5 - 25.1i)T^{2}
31 1+2.34T+31T2 1 + 2.34T + 31T^{2}
37 1+(−3.72+6.44i)T+(−18.5−32.0i)T2 1 + (-3.72 + 6.44i)T + (-18.5 - 32.0i)T^{2}
41 1+(0.425−0.736i)T+(−20.5−35.5i)T2 1 + (0.425 - 0.736i)T + (-20.5 - 35.5i)T^{2}
43 1+(−0.807−1.39i)T+(−21.5+37.2i)T2 1 + (-0.807 - 1.39i)T + (-21.5 + 37.2i)T^{2}
47 1+2.44T+47T2 1 + 2.44T + 47T^{2}
53 1+9.96T+53T2 1 + 9.96T + 53T^{2}
59 1+(2.69+4.66i)T+(−29.5+51.0i)T2 1 + (2.69 + 4.66i)T + (-29.5 + 51.0i)T^{2}
61 1+(−6.62−11.4i)T+(−30.5+52.8i)T2 1 + (-6.62 - 11.4i)T + (-30.5 + 52.8i)T^{2}
67 1+(−7.19+12.4i)T+(−33.5−58.0i)T2 1 + (-7.19 + 12.4i)T + (-33.5 - 58.0i)T^{2}
71 1+(−4.06−7.03i)T+(−35.5+61.4i)T2 1 + (-4.06 - 7.03i)T + (-35.5 + 61.4i)T^{2}
73 1−11.8T+73T2 1 - 11.8T + 73T^{2}
79 1−5.40T+79T2 1 - 5.40T + 79T^{2}
83 1+7.04T+83T2 1 + 7.04T + 83T^{2}
89 1+(−0.565+0.980i)T+(−44.5−77.0i)T2 1 + (-0.565 + 0.980i)T + (-44.5 - 77.0i)T^{2}
97 1+(2.97+5.14i)T+(−48.5+84.0i)T2 1 + (2.97 + 5.14i)T + (-48.5 + 84.0i)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

−11.01276402585211145902374166582, −9.706653370816498158113458103884, −8.646179140128976044275847462033, −7.57424861666838984228204096376, −6.82610553447482418659035332052, −5.00866379070956067825693471656, −4.28106571445404714140991828176, −3.33581714242086416098020702785, −2.34177057312061585192384103210, −0.54441248363324078977583531786, 3.28599081964485661315101016379, 3.93102796248669076083918032391, 4.88694565411790682803627033180, 5.85849630009398230571091395445, 6.94576455431698888455867938057, 7.983601551873093520705236610763, 8.191020834943332763886369205665, 9.311984255914000322317106038959, 10.81894683437230336068562494375, 11.51633713862091048501235344911

Graph of the ZZ-function along the critical line