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Properties

Label 2-507-13.9-c1-0-10
Degree 22
Conductor 507507
Sign 0.990+0.134i0.990 + 0.134i
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  + (−1.02 + 1.77i)2-s + (0.5 − 0.866i)3-s + (−1.09 − 1.90i)4-s − 3.35·5-s + (1.02 + 1.77i)6-s + (1.12 + 1.94i)7-s + 0.405·8-s + (−0.499 − 0.866i)9-s + (3.43 − 5.95i)10-s + (2.46 − 4.27i)11-s − 2.19·12-s − 4.60·14-s + (−1.67 + 2.90i)15-s + (1.78 − 3.08i)16-s + (−0.455 − 0.789i)17-s + 2.04·18-s + ⋯
L(s)  = 1  + (−0.724 + 1.25i)2-s + (0.288 − 0.499i)3-s + (−0.549 − 0.951i)4-s − 1.50·5-s + (0.418 + 0.724i)6-s + (0.424 + 0.735i)7-s + 0.143·8-s + (−0.166 − 0.288i)9-s + (1.08 − 1.88i)10-s + (0.744 − 1.28i)11-s − 0.634·12-s − 1.23·14-s + (−0.433 + 0.750i)15-s + (0.445 − 0.771i)16-s + (−0.110 − 0.191i)17-s + 0.482·18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((0.990+0.134i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((0.990+0.134i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: 0.990+0.134i0.990 + 0.134i
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), 0.990+0.134i)(2,\ 507,\ (\ :1/2),\ 0.990 + 0.134i)

Particular Values

L(1)L(1) ≈\approx 0.671220−0.0452133i0.671220 - 0.0452133i
L(12)L(\frac12) ≈\approx 0.671220−0.0452133i0.671220 - 0.0452133i
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+(1.02−1.77i)T+(−1−1.73i)T2 1 + (1.02 - 1.77i)T + (-1 - 1.73i)T^{2}
5 1+3.35T+5T2 1 + 3.35T + 5T^{2}
7 1+(−1.12−1.94i)T+(−3.5+6.06i)T2 1 + (-1.12 - 1.94i)T + (-3.5 + 6.06i)T^{2}
11 1+(−2.46+4.27i)T+(−5.5−9.52i)T2 1 + (-2.46 + 4.27i)T + (-5.5 - 9.52i)T^{2}
17 1+(0.455+0.789i)T+(−8.5+14.7i)T2 1 + (0.455 + 0.789i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.90+3.29i)T+(−9.5+16.4i)T2 1 + (1.90 + 3.29i)T + (-9.5 + 16.4i)T^{2}
23 1+(1.01−1.75i)T+(−11.5−19.9i)T2 1 + (1.01 - 1.75i)T + (-11.5 - 19.9i)T^{2}
29 1+(−1.96+3.41i)T+(−14.5−25.1i)T2 1 + (-1.96 + 3.41i)T + (-14.5 - 25.1i)T^{2}
31 1−8.82T+31T2 1 - 8.82T + 31T^{2}
37 1+(−4.40+7.62i)T+(−18.5−32.0i)T2 1 + (-4.40 + 7.62i)T + (-18.5 - 32.0i)T^{2}
41 1+(−3.46+6.00i)T+(−20.5−35.5i)T2 1 + (-3.46 + 6.00i)T + (-20.5 - 35.5i)T^{2}
43 1+(−1.14−1.97i)T+(−21.5+37.2i)T2 1 + (-1.14 - 1.97i)T + (-21.5 + 37.2i)T^{2}
47 1+3.80T+47T2 1 + 3.80T + 47T^{2}
53 1−0.542T+53T2 1 - 0.542T + 53T^{2}
59 1+(2.35+4.08i)T+(−29.5+51.0i)T2 1 + (2.35 + 4.08i)T + (-29.5 + 51.0i)T^{2}
61 1+(1.83+3.18i)T+(−30.5+52.8i)T2 1 + (1.83 + 3.18i)T + (-30.5 + 52.8i)T^{2}
67 1+(0.760−1.31i)T+(−33.5−58.0i)T2 1 + (0.760 - 1.31i)T + (-33.5 - 58.0i)T^{2}
71 1+(−1.18−2.05i)T+(−35.5+61.4i)T2 1 + (-1.18 - 2.05i)T + (-35.5 + 61.4i)T^{2}
73 1+7.41T+73T2 1 + 7.41T + 73T^{2}
79 1+3.74T+79T2 1 + 3.74T + 79T^{2}
83 1+2.30T+83T2 1 + 2.30T + 83T^{2}
89 1+(5.02−8.71i)T+(−44.5−77.0i)T2 1 + (5.02 - 8.71i)T + (-44.5 - 77.0i)T^{2}
97 1+(8.06+13.9i)T+(−48.5+84.0i)T2 1 + (8.06 + 13.9i)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

−11.11067515884928089367705741191, −9.453010777598746035722693181521, −8.555180631326469423040534602925, −8.289714381814471901917375906605, −7.43673493436967250653317489491, −6.56305332323564014638949561189, −5.68755344814861957342415173623, −4.24004163345415151061307309970, −2.91388535327318393574000253626, −0.56154658524569404100272218791, 1.32037328581597356234840187406, 2.91006810726655084707858498355, 4.12798171446516750194767231604, 4.43359241058474516446512670875, 6.60840691433536356160434783265, 7.81768680488412609086659710634, 8.322793297264157429834991597208, 9.376616877668326657782812872209, 10.23547113594437738078261786039, 10.77959086477612137515564685839

Graph of the ZZ-function along the critical line