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Properties

Label 2-507-39.8-c1-0-34
Degree 22
Conductor 507507
Sign 0.522+0.852i0.522 + 0.852i
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  + (0.249 − 0.249i)2-s + (0.892 − 1.48i)3-s + 1.87i·4-s + (2.45 − 2.45i)5-s + (−0.147 − 0.592i)6-s + (0.821 − 0.821i)7-s + (0.965 + 0.965i)8-s + (−1.40 − 2.64i)9-s − 1.22i·10-s + (1.32 + 1.32i)11-s + (2.78 + 1.67i)12-s − 0.409i·14-s + (−1.45 − 5.84i)15-s − 3.27·16-s − 5.90·17-s + (−1.01 − 0.309i)18-s + ⋯
L(s)  = 1  + (0.176 − 0.176i)2-s + (0.515 − 0.857i)3-s + 0.937i·4-s + (1.09 − 1.09i)5-s + (−0.0602 − 0.241i)6-s + (0.310 − 0.310i)7-s + (0.341 + 0.341i)8-s + (−0.469 − 0.882i)9-s − 0.387i·10-s + (0.399 + 0.399i)11-s + (0.803 + 0.483i)12-s − 0.109i·14-s + (−0.375 − 1.50i)15-s − 0.817·16-s − 1.43·17-s + (−0.238 − 0.0728i)18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((0.522+0.852i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((0.522+0.852i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: 0.522+0.852i0.522 + 0.852i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(437,⋅)\chi_{507} (437, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), 0.522+0.852i)(2,\ 507,\ (\ :1/2),\ 0.522 + 0.852i)

Particular Values

L(1)L(1) ≈\approx 1.93075−1.08184i1.93075 - 1.08184i
L(12)L(\frac12) ≈\approx 1.93075−1.08184i1.93075 - 1.08184i
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+(−0.892+1.48i)T 1 + (-0.892 + 1.48i)T
13 1 1
good2 1+(−0.249+0.249i)T−2iT2 1 + (-0.249 + 0.249i)T - 2iT^{2}
5 1+(−2.45+2.45i)T−5iT2 1 + (-2.45 + 2.45i)T - 5iT^{2}
7 1+(−0.821+0.821i)T−7iT2 1 + (-0.821 + 0.821i)T - 7iT^{2}
11 1+(−1.32−1.32i)T+11iT2 1 + (-1.32 - 1.32i)T + 11iT^{2}
17 1+5.90T+17T2 1 + 5.90T + 17T^{2}
19 1+(−3.48−3.48i)T+19iT2 1 + (-3.48 - 3.48i)T + 19iT^{2}
23 1−2.70T+23T2 1 - 2.70T + 23T^{2}
29 1−2.68iT−29T2 1 - 2.68iT - 29T^{2}
31 1+(3.22+3.22i)T+31iT2 1 + (3.22 + 3.22i)T + 31iT^{2}
37 1+(−1.52+1.52i)T−37iT2 1 + (-1.52 + 1.52i)T - 37iT^{2}
41 1+(−4.81+4.81i)T−41iT2 1 + (-4.81 + 4.81i)T - 41iT^{2}
43 1−5.55iT−43T2 1 - 5.55iT - 43T^{2}
47 1+(2.23+2.23i)T+47iT2 1 + (2.23 + 2.23i)T + 47iT^{2}
53 1−2.46iT−53T2 1 - 2.46iT - 53T^{2}
59 1+(7.07+7.07i)T+59iT2 1 + (7.07 + 7.07i)T + 59iT^{2}
61 1+2.66T+61T2 1 + 2.66T + 61T^{2}
67 1+(−4.81−4.81i)T+67iT2 1 + (-4.81 - 4.81i)T + 67iT^{2}
71 1+(−8.20+8.20i)T−71iT2 1 + (-8.20 + 8.20i)T - 71iT^{2}
73 1+(9.13−9.13i)T−73iT2 1 + (9.13 - 9.13i)T - 73iT^{2}
79 1−1.10T+79T2 1 - 1.10T + 79T^{2}
83 1+(4.58−4.58i)T−83iT2 1 + (4.58 - 4.58i)T - 83iT^{2}
89 1+(−3.02−3.02i)T+89iT2 1 + (-3.02 - 3.02i)T + 89iT^{2}
97 1+(−8.67−8.67i)T+97iT2 1 + (-8.67 - 8.67i)T + 97iT^{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

−10.98080040119103256305800543606, −9.427637657436993178635320847692, −9.032173509680807816597431853926, −8.095120238411072005417964555941, −7.26610585529591735822238850153, −6.23884075255281335040117822519, −4.99460471146372601503016729839, −3.90771181237077228541232163783, −2.44393233973171126895504893103, −1.44715389558276851262985334367, 1.99142087241178446791936900392, 3.00097329970844626791988792920, 4.52124513885321023082356485642, 5.45358357522140554570869082533, 6.30757791126557854454724896686, 7.18053737930903289332529443075, 8.777499847151895735081864397220, 9.392904450038145113098379422863, 10.11574232624752804658893381444, 10.98092122528453311240234056993

Graph of the ZZ-function along the critical line