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Properties

Label 1-507-507.71-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: $\chi_{507} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (0:\ ),\ 0.972 + 0.233i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9074476390 + 0.1073322351i\)
\(L(\frac12)\) \(\approx\) \(0.9074476390 + 0.1073322351i\)
\(L(1)\) \(\approx\) \(0.7636059707 + 0.1645958748i\)
\(L(1)\) \(\approx\) \(0.7636059707 + 0.1645958748i\)

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

−23.83468063011321536867066232755, −22.424482245611170029847208622594, −21.836960057342410356496579409072, −20.747442130373075902525949182899, −20.16910260409296996108431926873, −19.4768960504128770422598062994, −18.621629725011244389463163822, −17.5588132538598234340261873636, −17.10796059892226425489270105354, −16.025432510956184843578134734040, −15.19012453806338296387757741147, −14.07972817903318756040156915780, −12.870963246570530361738413545, −12.04099519990118525680321648183, −11.46172373583926534027780462873, −10.67473362769580325024154825627, −9.30707474740764428923266450894, −8.77670028378578168910728239120, −7.79700869304661677302071094059, −7.126822993321702772703254432996, −5.42765945391405199905102715437, −4.23133169162328907369602164281, −3.582263649444548040287894738, −1.977152423628236126729070401624, −1.121241406611651198304575093888, 0.7968700044967391983836348227, 2.15007830009129904373205189010, 3.7465792328556544868015466991, 4.76776767126575624925564200674, 5.91410399375246226604135710345, 7.03323309773028715815387299343, 7.62592751958755287030779200184, 8.507320233374549458897341033019, 9.43540607562241351789850626881, 10.58025474308674565582666570808, 11.34494194478428256692628949041, 11.996422383926200100277207900742, 13.80603455536015669665099592666, 14.43846035212325749001956412003, 15.04816134738034142676846427949, 16.14970790852953343589778953534, 16.68307820757221056385730241121, 17.9554931533752280816557386294, 18.27422092097459511826723000590, 19.34125314755402377748041547447, 20.029865102411427840705651424926, 20.86946880216363718456004349236, 22.37465617732032090252482235419, 22.833992526554024451060005444881, 23.91759035289956020078917176101

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