Properties

Label 1-507-507.134-r1-0-0
Degree $1$
Conductor $507$
Sign $-0.489 - 0.872i$
Analytic cond. $54.4847$
Root an. cond. $54.4847$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0402 − 0.999i)2-s + (−0.996 + 0.0804i)4-s + (−0.354 − 0.935i)5-s + (0.919 + 0.391i)7-s + (0.120 + 0.992i)8-s + (−0.919 + 0.391i)10-s + (−0.845 − 0.534i)11-s + (0.354 − 0.935i)14-s + (0.987 − 0.160i)16-s + (0.919 + 0.391i)17-s + (0.5 − 0.866i)19-s + (0.428 + 0.903i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.748 + 0.663i)25-s + ⋯
L(s)  = 1  + (−0.0402 − 0.999i)2-s + (−0.996 + 0.0804i)4-s + (−0.354 − 0.935i)5-s + (0.919 + 0.391i)7-s + (0.120 + 0.992i)8-s + (−0.919 + 0.391i)10-s + (−0.845 − 0.534i)11-s + (0.354 − 0.935i)14-s + (0.987 − 0.160i)16-s + (0.919 + 0.391i)17-s + (0.5 − 0.866i)19-s + (0.428 + 0.903i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.748 + 0.663i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.489 - 0.872i$
Analytic conductor: \(54.4847\)
Root analytic conductor: \(54.4847\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (1:\ ),\ -0.489 - 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8879900781 - 1.516267873i\)
\(L(\frac12)\) \(\approx\) \(0.8879900781 - 1.516267873i\)
\(L(1)\) \(\approx\) \(0.8315845923 - 0.6079336061i\)
\(L(1)\) \(\approx\) \(0.8315845923 - 0.6079336061i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.0402 - 0.999i)T \)
5 \( 1 + (-0.354 - 0.935i)T \)
7 \( 1 + (0.919 + 0.391i)T \)
11 \( 1 + (-0.845 - 0.534i)T \)
17 \( 1 + (0.919 + 0.391i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.0402 + 0.999i)T \)
31 \( 1 + (0.748 + 0.663i)T \)
37 \( 1 + (0.200 - 0.979i)T \)
41 \( 1 + (0.278 + 0.960i)T \)
43 \( 1 + (-0.200 - 0.979i)T \)
47 \( 1 + (0.568 - 0.822i)T \)
53 \( 1 + (-0.120 - 0.992i)T \)
59 \( 1 + (0.987 + 0.160i)T \)
61 \( 1 + (0.799 + 0.600i)T \)
67 \( 1 + (0.996 + 0.0804i)T \)
71 \( 1 + (0.692 - 0.721i)T \)
73 \( 1 + (-0.885 + 0.464i)T \)
79 \( 1 + (0.568 - 0.822i)T \)
83 \( 1 + (-0.970 + 0.239i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.632 + 0.774i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.59981800461775003123389345236, −23.01079560412834328221740570455, −22.40814770475820841599929437742, −21.1711281567390695282504789193, −20.47607113807971504160922979170, −18.951020766598179899995192169815, −18.56221356545911561734465524610, −17.69375291090940653024568520307, −16.91032576115467380788737809813, −15.85652925500875668319676367429, −15.10676671053554603168036530293, −14.376574634137009531801582726435, −13.77171491337220109945018141787, −12.55150195981233738100661564198, −11.47940101864435864051085229548, −10.367095678870775207992638535833, −9.75722140756596570273111760179, −8.1491730640286483342137908717, −7.782239878739914693207250445142, −6.93223918785449792225867511695, −5.83021125649449529033807871926, −4.82593135825881783391831573137, −3.91148015768479628425176631166, −2.60333807872409731233038480970, −0.88732682589383815833791328429, 0.64693968592562397509784247463, 1.55299816767956194504311055364, 2.823412834795690891729120916361, 3.898521031566465811188041163439, 5.19349605645867519297985228660, 5.33483184554799640000501438909, 7.512269726738952943572178975507, 8.373910328059276303190041135869, 8.9552357012490228355403077884, 10.07912759143867169449962559756, 11.12458799900579129422677766738, 11.74862089164012461919412803268, 12.6302533302439832970069330702, 13.38439848544187517300749601543, 14.32649571205153561768953013109, 15.39984760792323970441450981580, 16.40244082702442763689313342867, 17.427108113201019323040501497127, 18.11018914202167275679519477332, 19.0659957952530380488863236801, 19.80814761481438893162950157527, 20.72521621622514299611767033846, 21.28972135804024349066027265154, 21.88032489796990734772644243777, 23.339625188714741351730933554376

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