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Properties

Label 2-507-39.5-c1-0-25
Degree 22
Conductor 507507
Sign 0.957+0.289i0.957 + 0.289i
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  + 1.73·3-s − 2i·4-s + (3.09 + 3.09i)7-s + 2.99·9-s − 3.46i·12-s − 4·16-s + (−2.26 + 2.26i)19-s + (5.36 + 5.36i)21-s − 5i·25-s + 5.19·27-s + (6.19 − 6.19i)28-s + (0.830 − 0.830i)31-s − 5.99i·36-s + (−8.46 − 8.46i)37-s + 1.73i·43-s + ⋯
L(s)  = 1  + 1.00·3-s − i·4-s + (1.17 + 1.17i)7-s + 0.999·9-s − 0.999i·12-s − 16-s + (−0.520 + 0.520i)19-s + (1.17 + 1.17i)21-s − i·25-s + 1.00·27-s + (1.17 − 1.17i)28-s + (0.149 − 0.149i)31-s − 0.999i·36-s + (−1.39 − 1.39i)37-s + 0.264i·43-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((0.957+0.289i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((0.957+0.289i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: 0.957+0.289i0.957 + 0.289i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(239,⋅)\chi_{507} (239, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), 0.957+0.289i)(2,\ 507,\ (\ :1/2),\ 0.957 + 0.289i)

Particular Values

L(1)L(1) ≈\approx 2.17242−0.321668i2.17242 - 0.321668i
L(12)L(\frac12) ≈\approx 2.17242−0.321668i2.17242 - 0.321668i
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 1−1.73T 1 - 1.73T
13 1 1
good2 1+2iT2 1 + 2iT^{2}
5 1+5iT2 1 + 5iT^{2}
7 1+(−3.09−3.09i)T+7iT2 1 + (-3.09 - 3.09i)T + 7iT^{2}
11 1−11iT2 1 - 11iT^{2}
17 1+17T2 1 + 17T^{2}
19 1+(2.26−2.26i)T−19iT2 1 + (2.26 - 2.26i)T - 19iT^{2}
23 1+23T2 1 + 23T^{2}
29 1−29T2 1 - 29T^{2}
31 1+(−0.830+0.830i)T−31iT2 1 + (-0.830 + 0.830i)T - 31iT^{2}
37 1+(8.46+8.46i)T+37iT2 1 + (8.46 + 8.46i)T + 37iT^{2}
41 1+41iT2 1 + 41iT^{2}
43 1−1.73iT−43T2 1 - 1.73iT - 43T^{2}
47 1−47iT2 1 - 47iT^{2}
53 1−53T2 1 - 53T^{2}
59 1−59iT2 1 - 59iT^{2}
61 1−8.66T+61T2 1 - 8.66T + 61T^{2}
67 1+(11.5−11.5i)T−67iT2 1 + (11.5 - 11.5i)T - 67iT^{2}
71 1+71iT2 1 + 71iT^{2}
73 1+(7.63+7.63i)T+73iT2 1 + (7.63 + 7.63i)T + 73iT^{2}
79 1+12.1T+79T2 1 + 12.1T + 79T^{2}
83 1+83iT2 1 + 83iT^{2}
89 1−89iT2 1 - 89iT^{2}
97 1+(7.02−7.02i)T−97iT2 1 + (7.02 - 7.02i)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

−10.70231519726415158838410462051, −9.949215518502308110413263137817, −8.882016183464391243933687964247, −8.493162812494041066391399964481, −7.38318813135460189955454378731, −6.13600869747433017216839382978, −5.19014445140734830202037017848, −4.20998871526297960127435630745, −2.49641399468138284479520037360, −1.65867033055484333747565087492, 1.67722203884899974366815108604, 3.10626912948006536621911433562, 4.08615422428834798132064129461, 4.85577079397094602955820392245, 6.89921467456461448234756600202, 7.43921663760680981755768897117, 8.278336986794601750992838545987, 8.845174159931809766541751142262, 10.05911361376090806514910565905, 10.95296835000448805075020943361

Graph of the ZZ-function along the critical line