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Properties

Label 1-507-507.398-r0-0-0
Degree $1$
Conductor $507$
Sign $0.337 + 0.941i$
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  + (−0.663 + 0.748i)2-s + (−0.120 − 0.992i)4-s + (−0.239 + 0.970i)5-s + (−0.822 − 0.568i)7-s + (0.822 + 0.568i)8-s + (−0.568 − 0.822i)10-s + (−0.663 − 0.748i)11-s + (0.970 − 0.239i)14-s + (−0.970 + 0.239i)16-s + (0.568 − 0.822i)17-s + i·19-s + (0.992 + 0.120i)20-s + 22-s + 23-s + (−0.885 − 0.464i)25-s + ⋯
L(s)  = 1  + (−0.663 + 0.748i)2-s + (−0.120 − 0.992i)4-s + (−0.239 + 0.970i)5-s + (−0.822 − 0.568i)7-s + (0.822 + 0.568i)8-s + (−0.568 − 0.822i)10-s + (−0.663 − 0.748i)11-s + (0.970 − 0.239i)14-s + (−0.970 + 0.239i)16-s + (0.568 − 0.822i)17-s + i·19-s + (0.992 + 0.120i)20-s + 22-s + 23-s + (−0.885 − 0.464i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.337 + 0.941i$
Analytic conductor: \(2.35449\)
Root analytic conductor: \(2.35449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (398, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (0:\ ),\ 0.337 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5967233645 + 0.4201303762i\)
\(L(\frac12)\) \(\approx\) \(0.5967233645 + 0.4201303762i\)
\(L(1)\) \(\approx\) \(0.6273708860 + 0.2501939514i\)
\(L(1)\) \(\approx\) \(0.6273708860 + 0.2501939514i\)

Euler product

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

−23.310915652189284577435082276678, −22.514867143840728274732532446290, −21.42931631910346103584275740723, −20.9074830277268138635262597153, −19.950538832721022428415336175554, −19.31698198077481203681729448495, −18.57822162233540789741686000853, −17.4399567554512528445239589697, −16.89540431044711393712006001892, −15.8266385265594267767929827832, −15.27256399533512364956958469838, −13.470414329440714372829082609408, −12.8245716249891973299338518414, −12.2584300379199484712678697724, −11.3223842941446185280903194511, −10.1209790611749187650376351046, −9.48044519555460428211399186697, −8.603736897693073046885479570845, −7.8242967767701600814947432172, −6.66420318356704567602865911534, −5.2450615248390752114407670069, −4.2762588714035011657663381828, −3.08667838008542659126105726708, −2.091332361748800016758469831432, −0.70219094056547037296769148224, 0.87893625340171943157357068420, 2.67796026199863472731338542212, 3.636065145492750470475421123867, 5.15085547147333289981927446725, 6.15748698602465099975301349149, 6.99652827541567786962547922964, 7.65869420951597539090125386121, 8.70695089340018573905975853069, 9.86976333455597780850510250985, 10.4709148293847756059456240857, 11.2349799622284673665529332283, 12.633343531212849132036044709320, 13.94060665724746588216033131688, 14.24287835368162220723614549306, 15.52096534979457453456997690427, 16.07965023108376859761530154529, 16.854736887239281779211304068320, 17.923177482199446368679538827763, 18.76174210342161532183014180975, 19.16552286073280323335648391069, 20.12652263769020999338394846830, 21.25926258271136089789248646588, 22.44598784974498770340289217372, 23.22712305113017442835391777465, 23.51726536790067255923953350399

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