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Properties

Label 1-507-507.275-r0-0-0
Degree $1$
Conductor $507$
Sign $-0.699 - 0.714i$
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  + (−0.160 + 0.987i)2-s + (−0.948 − 0.316i)4-s + (−0.992 + 0.120i)5-s + (0.999 − 0.0402i)7-s + (0.464 − 0.885i)8-s + (0.0402 − 0.999i)10-s + (−0.774 + 0.632i)11-s + (−0.120 + 0.992i)14-s + (0.799 + 0.600i)16-s + (−0.0402 − 0.999i)17-s + (−0.866 + 0.5i)19-s + (0.979 + 0.200i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (0.970 − 0.239i)25-s + ⋯
L(s)  = 1  + (−0.160 + 0.987i)2-s + (−0.948 − 0.316i)4-s + (−0.992 + 0.120i)5-s + (0.999 − 0.0402i)7-s + (0.464 − 0.885i)8-s + (0.0402 − 0.999i)10-s + (−0.774 + 0.632i)11-s + (−0.120 + 0.992i)14-s + (0.799 + 0.600i)16-s + (−0.0402 − 0.999i)17-s + (−0.866 + 0.5i)19-s + (0.979 + 0.200i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (0.970 − 0.239i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.699 - 0.714i$
Analytic conductor: \(2.35449\)
Root analytic conductor: \(2.35449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (0:\ ),\ -0.699 - 0.714i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08292095395 + 0.1971585521i\)
\(L(\frac12)\) \(\approx\) \(-0.08292095395 + 0.1971585521i\)
\(L(1)\) \(\approx\) \(0.5258609785 + 0.3219481574i\)
\(L(1)\) \(\approx\) \(0.5258609785 + 0.3219481574i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.160 + 0.987i)T \)
5 \( 1 + (-0.992 + 0.120i)T \)
7 \( 1 + (0.999 - 0.0402i)T \)
11 \( 1 + (-0.774 + 0.632i)T \)
17 \( 1 + (-0.0402 - 0.999i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.987 - 0.160i)T \)
31 \( 1 + (-0.239 + 0.970i)T \)
37 \( 1 + (0.721 + 0.692i)T \)
41 \( 1 + (-0.903 + 0.428i)T \)
43 \( 1 + (-0.692 - 0.721i)T \)
47 \( 1 + (-0.663 + 0.748i)T \)
53 \( 1 + (-0.885 - 0.464i)T \)
59 \( 1 + (-0.600 - 0.799i)T \)
61 \( 1 + (-0.845 - 0.534i)T \)
67 \( 1 + (-0.316 - 0.948i)T \)
71 \( 1 + (0.0804 - 0.996i)T \)
73 \( 1 + (-0.935 - 0.354i)T \)
79 \( 1 + (-0.748 - 0.663i)T \)
83 \( 1 + (-0.822 + 0.568i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.391 + 0.919i)T \)
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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

−23.130205069347348728452937168534, −21.979227070977230047648691514992, −21.33677859247304184334062207915, −20.475218481472761146982847767339, −19.8259534460813574982494504066, −18.824515779095486407649856865101, −18.36831935778066710948517345313, −17.23908992188252816159735186245, −16.46477933694876075735642901089, −15.164860923174148497895795431993, −14.53799568338268860242876393254, −13.27476090414686748548670624671, −12.62169145684647887698968424777, −11.549880302596846557988654269164, −11.031731383989064014682346968125, −10.27502813416581885288121359035, −8.76883299125623270656773505073, −8.29379714523860254052941711046, −7.4884535871796041048565738601, −5.762566562699791369066483686386, −4.60752273593952126752570241554, −3.96751999979783328921614468860, −2.75623518172862263765880773032, −1.62972125017188814173959590051, −0.12381275175971832904851143318, 1.67692021014103193059310880901, 3.42056405608061841714903535836, 4.60740018651546309270466311483, 5.10851063378981106835858074404, 6.48281909876592470241104413011, 7.633716120069893729129909916653, 7.85962852820167182561398460608, 8.93157666201440886906616734535, 10.07723591813924561773491008707, 11.06066290728830948526787213529, 12.04379596920114264968560596045, 13.08448309697979821921672308145, 14.12699002283510124271538937703, 14.95843414030403569549818294074, 15.50121406398899965002236703144, 16.37840969151978522166570848930, 17.303002544102464179928070702736, 18.23268128762822718926717504121, 18.70549853246585367076618471767, 19.87403626094196154497454537879, 20.70059231843468076546432378267, 21.80281029478832792008098358127, 22.82797214570044351618525296476, 23.54462186726188225971434546388, 23.935406489694225204737680968322

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