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Properties

Label 2-507-507.362-c1-0-50
Degree $2$
Conductor $507$
Sign $-0.986 - 0.161i$
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  + (−1.70 + 0.277i)3-s + (0.783 − 1.83i)4-s + (−2.86 − 1.02i)7-s + (2.84 − 0.950i)9-s + (−0.829 + 3.36i)12-s + (−2.59 + 2.5i)13-s + (−2.77 − 2.88i)16-s + (−3.80 + 1.01i)19-s + (5.18 + 0.950i)21-s + (−2.32 + 4.42i)25-s + (−4.60 + 2.41i)27-s + (−4.12 + 4.47i)28-s + (−2.38 − 0.743i)31-s + (0.482 − 5.98i)36-s + (−8.40 + 1.89i)37-s + ⋯
L(s)  = 1  + (−0.987 + 0.160i)3-s + (0.391 − 0.919i)4-s + (−1.08 − 0.386i)7-s + (0.948 − 0.316i)9-s + (−0.239 + 0.970i)12-s + (−0.720 + 0.693i)13-s + (−0.692 − 0.721i)16-s + (−0.873 + 0.233i)19-s + (1.13 + 0.207i)21-s + (−0.464 + 0.885i)25-s + (−0.885 + 0.464i)27-s + (−0.780 + 0.845i)28-s + (−0.428 − 0.133i)31-s + (0.0804 − 0.996i)36-s + (−1.38 + 0.311i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.986 - 0.161i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.986 - 0.161i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0134287 + 0.165015i\)
\(L(\frac12)\) \(\approx\) \(0.0134287 + 0.165015i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 - 0.277i)T \)
13 \( 1 + (2.59 - 2.5i)T \)
good2 \( 1 + (-0.783 + 1.83i)T^{2} \)
5 \( 1 + (2.32 - 4.42i)T^{2} \)
7 \( 1 + (2.86 + 1.02i)T + (5.42 + 4.42i)T^{2} \)
11 \( 1 + (-6.60 + 8.79i)T^{2} \)
17 \( 1 + (-10.7 + 13.1i)T^{2} \)
19 \( 1 + (3.80 - 1.01i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (26.6 + 11.3i)T^{2} \)
31 \( 1 + (2.38 + 0.743i)T + (25.5 + 17.6i)T^{2} \)
37 \( 1 + (8.40 - 1.89i)T + (33.4 - 15.8i)T^{2} \)
41 \( 1 + (-12.9 - 38.8i)T^{2} \)
43 \( 1 + (6.24 + 9.87i)T + (-18.4 + 38.8i)T^{2} \)
47 \( 1 + (11.2 - 45.6i)T^{2} \)
53 \( 1 + (18.7 + 49.5i)T^{2} \)
59 \( 1 + (58.9 - 2.37i)T^{2} \)
61 \( 1 + (-13.9 + 1.12i)T + (60.2 - 9.78i)T^{2} \)
67 \( 1 + (7.70 + 3.09i)T + (48.3 + 46.4i)T^{2} \)
71 \( 1 + (69.5 + 14.2i)T^{2} \)
73 \( 1 + (0.620 - 10.2i)T + (-72.4 - 8.79i)T^{2} \)
79 \( 1 + (-0.946 + 7.79i)T + (-76.7 - 18.9i)T^{2} \)
83 \( 1 + (55.0 + 62.1i)T^{2} \)
89 \( 1 + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-14.8 + 8.20i)T + (51.8 - 81.9i)T^{2} \)
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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

−10.29600509883245830103737359845, −9.945212537507059626583740602556, −8.999274955206870122146992703501, −7.18500627370902933158780934146, −6.72448950940372618302530585446, −5.82450887285398375350877372508, −4.92198875657974748840061006882, −3.70336101829441430406498207923, −1.87266979316987965771908126603, −0.10246907074490054610449190995, 2.32542369187504560425345313685, 3.55267544386246654529625163901, 4.81158780027768143676343109924, 6.04315433277839692477957116193, 6.72398912002273232961233234039, 7.58549026278518607041518946759, 8.621401954691243837435498050931, 9.803218988123580608243310519726, 10.55109306026231288217971198613, 11.53064260626520222343211424006

Graph of the $Z$-function along the critical line