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Properties

Label 2-507-169.155-c1-0-26
Degree $2$
Conductor $507$
Sign $-0.580 + 0.814i$
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  + (−0.279 − 0.193i)2-s + (−0.885 + 0.464i)3-s + (−0.668 − 1.76i)4-s + (2.32 − 2.62i)5-s + (0.337 + 0.0409i)6-s + (0.713 − 2.89i)7-s + (−0.316 + 1.28i)8-s + (0.568 − 0.822i)9-s + (−1.15 + 0.284i)10-s + (1.95 − 1.34i)11-s + (1.41 + 1.24i)12-s + (−1.48 + 3.28i)13-s + (−0.758 + 0.672i)14-s + (−0.837 + 3.39i)15-s + (−2.48 + 2.20i)16-s + (−0.847 − 0.208i)17-s + ⋯
L(s)  = 1  + (−0.197 − 0.136i)2-s + (−0.511 + 0.268i)3-s + (−0.334 − 0.880i)4-s + (1.03 − 1.17i)5-s + (0.137 + 0.0167i)6-s + (0.269 − 1.09i)7-s + (−0.111 + 0.453i)8-s + (0.189 − 0.274i)9-s + (−0.365 + 0.0900i)10-s + (0.589 − 0.406i)11-s + (0.407 + 0.360i)12-s + (−0.413 + 0.910i)13-s + (−0.202 + 0.179i)14-s + (−0.216 + 0.877i)15-s + (−0.621 + 0.550i)16-s + (−0.205 − 0.0506i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.580 + 0.814i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.580 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.507406 - 0.985280i\)
\(L(\frac12)\) \(\approx\) \(0.507406 - 0.985280i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.885 - 0.464i)T \)
13 \( 1 + (1.48 - 3.28i)T \)
good2 \( 1 + (0.279 + 0.193i)T + (0.709 + 1.87i)T^{2} \)
5 \( 1 + (-2.32 + 2.62i)T + (-0.602 - 4.96i)T^{2} \)
7 \( 1 + (-0.713 + 2.89i)T + (-6.19 - 3.25i)T^{2} \)
11 \( 1 + (-1.95 + 1.34i)T + (3.90 - 10.2i)T^{2} \)
17 \( 1 + (0.847 + 0.208i)T + (15.0 + 7.90i)T^{2} \)
19 \( 1 + 1.22iT - 19T^{2} \)
23 \( 1 + 7.94T + 23T^{2} \)
29 \( 1 + (-2.54 + 3.69i)T + (-10.2 - 27.1i)T^{2} \)
31 \( 1 + (-9.65 - 1.17i)T + (30.0 + 7.41i)T^{2} \)
37 \( 1 + (-2.67 - 0.324i)T + (35.9 + 8.85i)T^{2} \)
41 \( 1 + (-1.53 - 2.92i)T + (-23.2 + 33.7i)T^{2} \)
43 \( 1 + (0.483 + 3.98i)T + (-41.7 + 10.2i)T^{2} \)
47 \( 1 + (7.35 + 2.78i)T + (35.1 + 31.1i)T^{2} \)
53 \( 1 + (5.25 + 1.29i)T + (46.9 + 24.6i)T^{2} \)
59 \( 1 + (4.76 - 5.37i)T + (-7.11 - 58.5i)T^{2} \)
61 \( 1 + (3.10 - 0.764i)T + (54.0 - 28.3i)T^{2} \)
67 \( 1 + (-0.740 - 0.280i)T + (50.1 + 44.4i)T^{2} \)
71 \( 1 + (2.16 + 4.13i)T + (-40.3 + 58.4i)T^{2} \)
73 \( 1 + (-7.38 + 5.09i)T + (25.8 - 68.2i)T^{2} \)
79 \( 1 + (5.73 - 15.1i)T + (-59.1 - 52.3i)T^{2} \)
83 \( 1 + (-4.07 + 7.75i)T + (-47.1 - 68.3i)T^{2} \)
89 \( 1 + 8.25iT - 89T^{2} \)
97 \( 1 + (-11.4 - 12.9i)T + (-11.6 + 96.2i)T^{2} \)
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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

−10.31030921149842322097743488503, −9.818151531908250350549164104170, −9.131443856397944943479210016407, −8.166060329701760783715530035078, −6.55669577339932718565702186265, −5.93529052557002920532739993638, −4.74365007326647141465529680356, −4.33506608426853217416822819689, −1.84371534527738946271941494836, −0.77905667139309367640653764842, 2.13064353219255098895048632813, 3.10099278141045758734672901995, 4.68620413409552000175137493149, 5.94924563616559239185658143554, 6.50259196283395482640764449077, 7.61220320015826818503164497029, 8.435231116708286596229931225595, 9.620540786360246700063613645148, 10.14409523764185346894723123112, 11.31339452445834797397543716317

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