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Properties

Label 1-507-507.317-r0-0-0
Degree $1$
Conductor $507$
Sign $0.690 - 0.723i$
Analytic cond. $2.35449$
Root an. cond. $2.35449$
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  + (−0.992 + 0.120i)2-s + (0.970 − 0.239i)4-s + (−0.464 + 0.885i)5-s + (0.935 − 0.354i)7-s + (−0.935 + 0.354i)8-s + (0.354 − 0.935i)10-s + (−0.992 − 0.120i)11-s + (−0.885 + 0.464i)14-s + (0.885 − 0.464i)16-s + (−0.354 − 0.935i)17-s − i·19-s + (−0.239 + 0.970i)20-s + 22-s + 23-s + (−0.568 − 0.822i)25-s + ⋯
L(s)  = 1  + (−0.992 + 0.120i)2-s + (0.970 − 0.239i)4-s + (−0.464 + 0.885i)5-s + (0.935 − 0.354i)7-s + (−0.935 + 0.354i)8-s + (0.354 − 0.935i)10-s + (−0.992 − 0.120i)11-s + (−0.885 + 0.464i)14-s + (0.885 − 0.464i)16-s + (−0.354 − 0.935i)17-s − i·19-s + (−0.239 + 0.970i)20-s + 22-s + 23-s + (−0.568 − 0.822i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.690 - 0.723i$
Analytic conductor: \(2.35449\)
Root analytic conductor: \(2.35449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (0:\ ),\ 0.690 - 0.723i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6411153831 - 0.2743279558i\)
\(L(\frac12)\) \(\approx\) \(0.6411153831 - 0.2743279558i\)
\(L(1)\) \(\approx\) \(0.6616854291 + 0.02019362527i\)
\(L(1)\) \(\approx\) \(0.6616854291 + 0.02019362527i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.992 + 0.120i)T \)
5 \( 1 + (-0.464 + 0.885i)T \)
7 \( 1 + (0.935 - 0.354i)T \)
11 \( 1 + (-0.992 - 0.120i)T \)
17 \( 1 + (-0.354 - 0.935i)T \)
19 \( 1 - iT \)
23 \( 1 + T \)
29 \( 1 + (-0.120 - 0.992i)T \)
31 \( 1 + (-0.822 - 0.568i)T \)
37 \( 1 + (0.822 + 0.568i)T \)
41 \( 1 + (0.663 - 0.748i)T \)
43 \( 1 + (-0.568 - 0.822i)T \)
47 \( 1 + (-0.239 + 0.970i)T \)
53 \( 1 + (0.354 + 0.935i)T \)
59 \( 1 + (0.464 - 0.885i)T \)
61 \( 1 + (-0.354 + 0.935i)T \)
67 \( 1 + (-0.239 + 0.970i)T \)
71 \( 1 + (0.663 - 0.748i)T \)
73 \( 1 + (0.992 + 0.120i)T \)
79 \( 1 + (-0.970 - 0.239i)T \)
83 \( 1 + (-0.663 - 0.748i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.464 - 0.885i)T \)
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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

−23.9195962926577902162911598574, −23.203540728687890611305461441963, −21.45306381976338642539954958566, −21.16983276525911173219785724405, −20.21560846338100771622415744879, −19.57624964016435452846132939166, −18.46033170991910866174241786027, −17.95455126987854014473477527370, −16.8866878956114110381312131667, −16.29637663950184812734540514410, −15.30565653001931284508775039208, −14.65850859700662522809273346865, −12.9511116181679428527614529106, −12.43659818285406746917632763873, −11.32718011086022568465899555357, −10.72980982816682536548003415472, −9.56649130581559632389170355759, −8.53453925679618599715858552047, −8.12658414633019624928179683983, −7.20710887969053012753168852896, −5.76672760303988466535625699870, −4.87283116294321844047956903369, −3.53897316797274651898263428372, −2.130513776388386825895401730755, −1.20988406056618772974271524675, 0.57112895527701923937745529183, 2.21396994491229774547903828845, 2.98942972140168159540688203693, 4.53718154877771395573475403571, 5.721259730343915841744149369186, 7.0486105325261206554794054693, 7.4882022734114300032215489665, 8.37726946890315804848185183692, 9.45012386795050405988458103539, 10.548128978288109630842729589245, 11.14370953089433368458543517810, 11.700480165613621899045277779049, 13.22024400627977272448393549095, 14.29888152697747699798132098086, 15.232330599071291034999765354675, 15.706760420736925194230089989561, 16.874963318748228991712978591328, 17.7447498020155135817704158106, 18.38709602771800834707965208277, 19.03925340175750610082159736164, 20.07376888548718419185173445556, 20.75872920517358099654540517046, 21.6323049066987838931779393357, 22.82598873152847169992847327273, 23.74236751746059880420552025606

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