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Properties

Label 2-507-39.5-c1-0-25
Degree $2$
Conductor $507$
Sign $0.957 + 0.289i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.957 + 0.289i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17242 - 0.321668i\)
\(L(\frac12)\) \(\approx\) \(2.17242 - 0.321668i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
13 \( 1 \)
good2 \( 1 + 2iT^{2} \)
5 \( 1 + 5iT^{2} \)
7 \( 1 + (-3.09 - 3.09i)T + 7iT^{2} \)
11 \( 1 - 11iT^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (2.26 - 2.26i)T - 19iT^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-0.830 + 0.830i)T - 31iT^{2} \)
37 \( 1 + (8.46 + 8.46i)T + 37iT^{2} \)
41 \( 1 + 41iT^{2} \)
43 \( 1 - 1.73iT - 43T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 59iT^{2} \)
61 \( 1 - 8.66T + 61T^{2} \)
67 \( 1 + (11.5 - 11.5i)T - 67iT^{2} \)
71 \( 1 + 71iT^{2} \)
73 \( 1 + (7.63 + 7.63i)T + 73iT^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 - 89iT^{2} \)
97 \( 1 + (7.02 - 7.02i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line