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Properties

Label 2-507-39.32-c1-0-34
Degree $2$
Conductor $507$
Sign $-0.624 - 0.781i$
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  + (−0.965 + 0.258i)2-s + (−0.724 − 1.57i)3-s + (−0.866 + 0.5i)4-s + (−1.41 − 1.41i)5-s + (1.10 + 1.33i)6-s + (0.366 − 1.36i)7-s + (2.12 − 2.12i)8-s + (−1.94 + 2.28i)9-s + (1.73 + 1.00i)10-s + (−1.03 − 3.86i)11-s + (1.41 + i)12-s + 1.41i·14-s + (−1.19 + 3.24i)15-s + (−0.500 + 0.866i)16-s + (1.29 − 2.70i)18-s + (−1.36 − 0.366i)19-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.418 − 0.908i)3-s + (−0.433 + 0.250i)4-s + (−0.632 − 0.632i)5-s + (0.452 + 0.543i)6-s + (0.138 − 0.516i)7-s + (0.749 − 0.749i)8-s + (−0.649 + 0.760i)9-s + (0.547 + 0.316i)10-s + (−0.312 − 1.16i)11-s + (0.408 + 0.288i)12-s + 0.377i·14-s + (−0.309 + 0.839i)15-s + (−0.125 + 0.216i)16-s + (0.304 − 0.638i)18-s + (−0.313 − 0.0839i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.624 - 0.781i$
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: \((2,\ 507,\ (\ :1/2),\ -0.624 - 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0273714 + 0.0569058i\)
\(L(\frac12)\) \(\approx\) \(0.0273714 + 0.0569058i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.724 + 1.57i)T \)
13 \( 1 \)
good2 \( 1 + (0.965 - 0.258i)T + (1.73 - i)T^{2} \)
5 \( 1 + (1.41 + 1.41i)T + 5iT^{2} \)
7 \( 1 + (-0.366 + 1.36i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.03 + 3.86i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.36 + 0.366i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.24 - 7.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.44 - 1.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5 - 5i)T - 31iT^{2} \)
37 \( 1 + (1.36 - 0.366i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.93 + 0.517i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.19 - 3i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.82 - 2.82i)T - 47iT^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 + (-3.86 - 1.03i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.83 + 6.83i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.03 + 3.86i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1 - i)T + 73iT^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + (5.65 + 5.65i)T + 83iT^{2} \)
89 \( 1 + (-3.62 - 13.5i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (9.56 + 2.56i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.41520610610336682609408087316, −9.160300945600301806706067974361, −8.286988747053848754555233961404, −7.86433282472590207839596801643, −6.98216380851729768758765375212, −5.74807928303966528559778262311, −4.63864725332895118203073143125, −3.43083137562711586876908852936, −1.29659564951024181329522911681, −0.05535976487160293539049137089, 2.31292254736253142222025339398, 3.94772449596502164374740914076, 4.76189096386381570070946140902, 5.77998429149338184176322959496, 7.05382700897029388551099623286, 8.186329743371337523741572954497, 8.928149812608829326458734177826, 9.973788201877052044253622007819, 10.32598695781925843592828972517, 11.25015493110933424640176698952

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