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Properties

Label 1-507-507.119-r0-0-0
Degree $1$
Conductor $507$
Sign $-0.972 + 0.233i$
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  + (0.721 + 0.692i)2-s + (0.0402 + 0.999i)4-s + (0.822 + 0.568i)5-s + (−0.979 − 0.200i)7-s + (−0.663 + 0.748i)8-s + (0.200 + 0.979i)10-s + (−0.960 − 0.278i)11-s + (−0.568 − 0.822i)14-s + (−0.996 + 0.0804i)16-s + (−0.200 + 0.979i)17-s + (−0.866 + 0.5i)19-s + (−0.534 + 0.845i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (0.354 + 0.935i)25-s + ⋯
L(s)  = 1  + (0.721 + 0.692i)2-s + (0.0402 + 0.999i)4-s + (0.822 + 0.568i)5-s + (−0.979 − 0.200i)7-s + (−0.663 + 0.748i)8-s + (0.200 + 0.979i)10-s + (−0.960 − 0.278i)11-s + (−0.568 − 0.822i)14-s + (−0.996 + 0.0804i)16-s + (−0.200 + 0.979i)17-s + (−0.866 + 0.5i)19-s + (−0.534 + 0.845i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (0.354 + 0.935i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.972 + 0.233i$
Analytic conductor: \(2.35449\)
Root analytic conductor: \(2.35449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (0:\ ),\ -0.972 + 0.233i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1698003723 + 1.435588728i\)
\(L(\frac12)\) \(\approx\) \(0.1698003723 + 1.435588728i\)
\(L(1)\) \(\approx\) \(0.9801780045 + 0.8579666863i\)
\(L(1)\) \(\approx\) \(0.9801780045 + 0.8579666863i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.721 + 0.692i)T \)
5 \( 1 + (0.822 + 0.568i)T \)
7 \( 1 + (-0.979 - 0.200i)T \)
11 \( 1 + (-0.960 - 0.278i)T \)
17 \( 1 + (-0.200 + 0.979i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.692 + 0.721i)T \)
31 \( 1 + (0.935 + 0.354i)T \)
37 \( 1 + (0.774 - 0.632i)T \)
41 \( 1 + (-0.600 + 0.799i)T \)
43 \( 1 + (0.632 - 0.774i)T \)
47 \( 1 + (-0.464 - 0.885i)T \)
53 \( 1 + (0.748 + 0.663i)T \)
59 \( 1 + (-0.0804 + 0.996i)T \)
61 \( 1 + (0.948 + 0.316i)T \)
67 \( 1 + (0.999 + 0.0402i)T \)
71 \( 1 + (-0.391 - 0.919i)T \)
73 \( 1 + (-0.239 - 0.970i)T \)
79 \( 1 + (0.885 - 0.464i)T \)
83 \( 1 + (-0.992 + 0.120i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.903 - 0.428i)T \)
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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

−22.990197931089291013568749058910, −22.35498916850484517680903902640, −21.51345590103569433571272717351, −20.714309984679157658351105736873, −20.17395264583387042075105361479, −19.05622775726677659563054871649, −18.39576351082892622890205246974, −17.34228200190729108626825593304, −16.16180748568762201523665452928, −15.53331926523742915953082826371, −14.408377336129627840919890521935, −13.35124102257841270818958867943, −13.02842365219705414122797370799, −12.19288271547305082249525805310, −11.066238979809865421731603483142, −9.95903714938252750494558106864, −9.60816664051009191979599592967, −8.42718363729922995668206176270, −6.78844425706497247707987004327, −6.005391777524617878506539010807, −5.083933080470268897454535746655, −4.215270293093635730966376678948, −2.72985219152433186652406781436, −2.238470762692438751740318438224, −0.5562133193530109496770628490, 2.114540296546213009378594937539, 3.11732777890502825132521307565, 4.005136648865754940482470200451, 5.44150371880758554518173621726, 6.07923436881940104756304265882, 6.85858821501009383521990734991, 7.873709854623838167628391249029, 8.95332885868545260494351682590, 10.10996511478331219645509650444, 10.84803305233851808781807766486, 12.22748884907108374343252722812, 13.1911833829057833014546580567, 13.49232874546464760166786757779, 14.6311250519063941123224846721, 15.34343020630125046314118062033, 16.294621308417042914230346556955, 17.02850608436616936935893673561, 17.91095633378705369717920831856, 18.780724821389076975300315736132, 19.85111690354003858791687434695, 21.072483587334579479504930447329, 21.622310231538887304877347754595, 22.34187367689658000716034554278, 23.248768145457831202615254717464, 23.79124587314411905899409418157

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