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Properties

Label 1-507-507.2-r0-0-0
Degree $1$
Conductor $507$
Sign $0.872 + 0.489i$
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  + (−0.999 − 0.0402i)2-s + (0.996 + 0.0804i)4-s + (−0.935 − 0.354i)5-s + (−0.391 − 0.919i)7-s + (−0.992 − 0.120i)8-s + (0.919 + 0.391i)10-s + (0.534 + 0.845i)11-s + (0.354 + 0.935i)14-s + (0.987 + 0.160i)16-s + (−0.919 + 0.391i)17-s + (−0.866 + 0.5i)19-s + (−0.903 − 0.428i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (0.748 + 0.663i)25-s + ⋯
L(s)  = 1  + (−0.999 − 0.0402i)2-s + (0.996 + 0.0804i)4-s + (−0.935 − 0.354i)5-s + (−0.391 − 0.919i)7-s + (−0.992 − 0.120i)8-s + (0.919 + 0.391i)10-s + (0.534 + 0.845i)11-s + (0.354 + 0.935i)14-s + (0.987 + 0.160i)16-s + (−0.919 + 0.391i)17-s + (−0.866 + 0.5i)19-s + (−0.903 − 0.428i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (0.748 + 0.663i)25-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.872 + 0.489i$
Analytic conductor: \(2.35449\)
Root analytic conductor: \(2.35449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (0:\ ),\ 0.872 + 0.489i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5317208921 + 0.1389486638i\)
\(L(\frac12)\) \(\approx\) \(0.5317208921 + 0.1389486638i\)
\(L(1)\) \(\approx\) \(0.5589503265 + 0.01007767007i\)
\(L(1)\) \(\approx\) \(0.5589503265 + 0.01007767007i\)

Euler product

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

−23.90312837523941119146502140904, −22.55688453159941496958561192011, −21.914508923906125439524138442252, −20.85174946121103885467776641442, −19.723266680925825315299630211919, −19.344456149032247116638768857855, −18.525803050833315309967958277326, −17.8339112080715504934731765820, −16.57171003742244552350156906279, −16.02201659503452086188292995631, −15.2078222212763626785439142507, −14.48395124495665746364909896893, −12.932790089020682700097406587476, −11.94841625624836071050807057461, −11.293933214093945757015482009001, −10.53245389347615839366402903405, −9.19905984323667159130343457627, −8.68138280477757957527786674475, −7.7779966445509189021987226, −6.61423669367223765622667073960, −6.08392488355776807001691326172, −4.42263641493483313288700391851, −3.099865084099281345462768259935, −2.31430780886595869799951320679, −0.54337000441276241363345939938, 0.917545682395888770841381554671, 2.1908311948517162875715706682, 3.70808061265326916764368246699, 4.3812701664993184840444954083, 6.147137138244521425457473857476, 7.057590440988323826671471966834, 7.76962773693673348981073023310, 8.68281480603260479249054887056, 9.6492541904601522849728781913, 10.50133327943773490325354431401, 11.389079265461142754662636592960, 12.241224181573367390166017902, 13.08725160653750059732377080467, 14.48976945739393698688417801383, 15.47440504081471185531590241282, 16.03951028300084340608567239632, 17.16727586994284092077492708963, 17.4201290985501553625748199883, 18.79464836023502813639377037572, 19.57246164029874582183159978675, 20.02324288114839069892838557574, 20.7435914594360112985711410651, 21.923404610725914380327295000474, 23.17392127071204483703587131513, 23.607134286810508153663835317226

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