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Properties

Label 2-507-13.3-c1-0-4
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(484,⋅)\chi_{507} (484, \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

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

Graph of the ZZ-function along the critical line