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Properties

Label 1-507-507.113-r1-0-0
Degree $1$
Conductor $507$
Sign $-0.688 - 0.725i$
Analytic cond. $54.4847$
Root an. cond. $54.4847$
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  + (−0.278 − 0.960i)2-s + (−0.845 + 0.534i)4-s + (−0.568 + 0.822i)5-s + (0.948 − 0.316i)7-s + (0.748 + 0.663i)8-s + (0.948 + 0.316i)10-s + (−0.692 − 0.721i)11-s + (−0.568 − 0.822i)14-s + (0.428 − 0.903i)16-s + (−0.948 + 0.316i)17-s + (−0.5 + 0.866i)19-s + (0.0402 − 0.999i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.354 − 0.935i)25-s + ⋯
L(s)  = 1  + (−0.278 − 0.960i)2-s + (−0.845 + 0.534i)4-s + (−0.568 + 0.822i)5-s + (0.948 − 0.316i)7-s + (0.748 + 0.663i)8-s + (0.948 + 0.316i)10-s + (−0.692 − 0.721i)11-s + (−0.568 − 0.822i)14-s + (0.428 − 0.903i)16-s + (−0.948 + 0.316i)17-s + (−0.5 + 0.866i)19-s + (0.0402 − 0.999i)20-s + (−0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.354 − 0.935i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.688 - 0.725i$
Analytic conductor: \(54.4847\)
Root analytic conductor: \(54.4847\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (1:\ ),\ -0.688 - 0.725i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3223257255 - 0.7500732545i\)
\(L(\frac12)\) \(\approx\) \(0.3223257255 - 0.7500732545i\)
\(L(1)\) \(\approx\) \(0.6941152715 - 0.2744353120i\)
\(L(1)\) \(\approx\) \(0.6941152715 - 0.2744353120i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.278 - 0.960i)T \)
5 \( 1 + (-0.568 + 0.822i)T \)
7 \( 1 + (0.948 - 0.316i)T \)
11 \( 1 + (-0.692 - 0.721i)T \)
17 \( 1 + (-0.948 + 0.316i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.278 - 0.960i)T \)
31 \( 1 + (-0.354 + 0.935i)T \)
37 \( 1 + (0.987 - 0.160i)T \)
41 \( 1 + (0.919 - 0.391i)T \)
43 \( 1 + (0.987 + 0.160i)T \)
47 \( 1 + (-0.885 + 0.464i)T \)
53 \( 1 + (0.748 + 0.663i)T \)
59 \( 1 + (-0.428 - 0.903i)T \)
61 \( 1 + (-0.200 - 0.979i)T \)
67 \( 1 + (-0.845 - 0.534i)T \)
71 \( 1 + (-0.799 - 0.600i)T \)
73 \( 1 + (-0.970 + 0.239i)T \)
79 \( 1 + (0.885 - 0.464i)T \)
83 \( 1 + (-0.120 - 0.992i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.996 + 0.0804i)T \)
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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

−24.02612748372068177854542532658, −23.167269873970511364795880135142, −22.2581418004962589394511380562, −21.12958980542268356825384286951, −20.28112542218493282739082361343, −19.44304617359513660527956492767, −18.280520482927035888745043006822, −17.79266548739690164060485344808, −16.830789882451938860180045508716, −16.040038572440030837283672927950, −15.14368597794819388158970176610, −14.7247777187771316227798597415, −13.30374723916063802159410475359, −12.77692278084244546650973794783, −11.49324373532338388549575599535, −10.59041801201008877278196084751, −9.2208525294104210514190203167, −8.68509658663376173103478255477, −7.76935982761615682655392749050, −7.05124107347254406891753811383, −5.69607449317937113851781976482, −4.71947545966117567519933927288, −4.36043096853094933338554854505, −2.317453963937058494816546851814, −0.91996572703028614607981180892, 0.28964107994529509893013706437, 1.71081721216010705707636788676, 2.77304517402026055937698720289, 3.79813822463675849161049193172, 4.63570037398442189958859029150, 5.96370682656581999047687796136, 7.50514931054775593007108224556, 8.020195607389275120853930186325, 9.01753055613131500632256029028, 10.319304331241451543521936197730, 10.99331749530777275701900878668, 11.40964850663040110579023799648, 12.55807761696822729481579460689, 13.56052389434174316551480774119, 14.33831780202596987496745194445, 15.2555554477010840739110929997, 16.397362942774738719857555755074, 17.53027275441114976167340555472, 18.09286890254543647492101448165, 19.011512098629381946139568841052, 19.56773312293664337625786845438, 20.62219006297019359266429877151, 21.33829996838780230034762978671, 22.01829423014213157570814092803, 23.1324933319133949465914960787

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