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Properties

Label 1-507-507.137-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  + (0.979 + 0.200i)2-s + (0.919 + 0.391i)4-s + (−0.239 + 0.970i)5-s + (0.903 − 0.428i)7-s + (0.822 + 0.568i)8-s + (−0.428 + 0.903i)10-s + (−0.316 + 0.948i)11-s + (0.970 − 0.239i)14-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + (−0.866 − 0.5i)19-s + (−0.600 + 0.799i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.885 − 0.464i)25-s + ⋯
L(s)  = 1  + (0.979 + 0.200i)2-s + (0.919 + 0.391i)4-s + (−0.239 + 0.970i)5-s + (0.903 − 0.428i)7-s + (0.822 + 0.568i)8-s + (−0.428 + 0.903i)10-s + (−0.316 + 0.948i)11-s + (0.970 − 0.239i)14-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + (−0.866 − 0.5i)19-s + (−0.600 + 0.799i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)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: $\chi_{507} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (0:\ ),\ 0.337 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.153074023 + 1.515898075i\)
\(L(\frac12)\) \(\approx\) \(2.153074023 + 1.515898075i\)
\(L(1)\) \(\approx\) \(1.827107452 + 0.6799489092i\)
\(L(1)\) \(\approx\) \(1.827107452 + 0.6799489092i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.979 + 0.200i)T \)
5 \( 1 + (-0.239 + 0.970i)T \)
7 \( 1 + (0.903 - 0.428i)T \)
11 \( 1 + (-0.316 + 0.948i)T \)
17 \( 1 + (0.428 + 0.903i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.200 - 0.979i)T \)
31 \( 1 + (0.464 + 0.885i)T \)
37 \( 1 + (-0.534 + 0.845i)T \)
41 \( 1 + (-0.160 - 0.987i)T \)
43 \( 1 + (0.845 - 0.534i)T \)
47 \( 1 + (0.992 + 0.120i)T \)
53 \( 1 + (-0.568 + 0.822i)T \)
59 \( 1 + (0.721 + 0.692i)T \)
61 \( 1 + (-0.996 + 0.0804i)T \)
67 \( 1 + (-0.391 - 0.919i)T \)
71 \( 1 + (-0.774 + 0.632i)T \)
73 \( 1 + (0.663 + 0.748i)T \)
79 \( 1 + (0.120 - 0.992i)T \)
83 \( 1 + (-0.935 + 0.354i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.960 + 0.278i)T \)
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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

−23.696920713562444485449135785652, −22.63447922742654304159492700836, −21.48851370839255587795400997934, −21.13114820852317258249486613105, −20.367389762417488590264971731826, −19.42667312889965503371156720092, −18.57035790859400964190624303733, −17.3334346600105573170157382037, −16.31653566964245328011475137621, −15.782202262051342692761541078054, −14.724978629329307369486073446584, −13.93447729217497261249745963974, −13.09024655349848321856076348708, −12.168928977913844726925997766093, −11.54430200515323567144073925583, −10.693413296266781088289557776166, −9.39011177569658716831472361583, −8.28240656227680349933377744670, −7.547192894480748451332817141149, −6.02228071703524204156948225279, −5.33140716686304581663114374702, −4.53210722402506365649433392807, −3.48970203300301504007177861122, −2.22381721828710498151105000228, −1.109526076017189188103087349339, 1.82773179161288890205382597670, 2.70635659305841276684405674991, 4.01733743366570891163467114675, 4.59958707627726503848406476841, 5.87549945189209824392959185508, 6.83197407854786095672439613389, 7.577977008067619831350684402096, 8.41647405819390994320250693818, 10.41322996301118073666082537869, 10.60504805096931800434789667214, 11.815960914008261774101524064549, 12.48719380749980536407997588942, 13.68937066043814421908967737252, 14.36657306719581149466266519015, 15.11728924954421046962031033949, 15.649944513331414764671006297539, 17.09254287439115334307512042602, 17.57427154058039385998475533165, 18.78831932364562989254661084755, 19.73315528456178532730780965609, 20.65863838190732196130774975884, 21.33189315406487474505219102144, 22.21692325370920841924387609177, 23.02264227555524648098351587530, 23.63065342407697850774936819091

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