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Properties

Label 2-507-39.32-c1-0-3
Degree $2$
Conductor $507$
Sign $-0.525 - 0.850i$
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  + (−2.31 + 0.619i)2-s + (−1.28 − 1.16i)3-s + (3.23 − 1.86i)4-s + (1.69 + 1.69i)5-s + (3.68 + 1.89i)6-s + (−0.366 + 1.36i)7-s + (−2.93 + 2.93i)8-s + (0.292 + 2.98i)9-s + (−4.96 − 2.86i)10-s + (0.453 + 1.69i)11-s + (−6.31 − 1.36i)12-s − 3.38i·14-s + (−0.202 − 4.14i)15-s + (1.23 − 2.13i)16-s + (−1.07 − 1.85i)17-s + (−2.52 − 6.72i)18-s + ⋯
L(s)  = 1  + (−1.63 + 0.438i)2-s + (−0.740 − 0.671i)3-s + (1.61 − 0.933i)4-s + (0.757 + 0.757i)5-s + (1.50 + 0.773i)6-s + (−0.138 + 0.516i)7-s + (−1.03 + 1.03i)8-s + (0.0975 + 0.995i)9-s + (−1.56 − 0.906i)10-s + (0.136 + 0.510i)11-s + (−1.82 − 0.394i)12-s − 0.904i·14-s + (−0.0523 − 1.06i)15-s + (0.308 − 0.533i)16-s + (−0.260 − 0.450i)17-s + (−0.595 − 1.58i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.525 - 0.850i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.525 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.189933 + 0.340471i\)
\(L(\frac12)\) \(\approx\) \(0.189933 + 0.340471i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.28 + 1.16i)T \)
13 \( 1 \)
good2 \( 1 + (2.31 - 0.619i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-1.69 - 1.69i)T + 5iT^{2} \)
7 \( 1 + (0.366 - 1.36i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.453 - 1.69i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.07 + 1.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 + 0.267i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.79 - 2.76i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.46 - 4.46i)T - 31iT^{2} \)
37 \( 1 + (6.59 - 1.76i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.619 + 0.166i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (7.09 - 4.09i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.77 - 6.77i)T - 47iT^{2} \)
53 \( 1 - 4.62iT - 53T^{2} \)
59 \( 1 + (4.62 + 1.23i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.26 + 8.46i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.23 - 4.62i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-6.09 - 6.09i)T + 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (1.23 + 1.23i)T + 83iT^{2} \)
89 \( 1 + (-2.60 - 9.70i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.5 - 3.36i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.81736019760469703247840438150, −10.30525091489613232160339112414, −9.438951080179929536661084673775, −8.547300177109661910738892062711, −7.51719472401874234307369767991, −6.69578645119004195849353253279, −6.28236745252107512935415501859, −5.08036488845456704348509514269, −2.57997850292195011871378432332, −1.51203502726482736753009556803, 0.44291226471568165609263552419, 1.78922642333466199372968203516, 3.59245078878281311721689405111, 4.99660342515332078245414627439, 6.11297347199183359659994637123, 7.07797278926254627525036526863, 8.408242576845092178992593456182, 8.996647286356543703998493178498, 9.864357868007408933363825777658, 10.32852594033869086054257820588

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