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Properties

Label 1-507-507.383-r0-0-0
Degree $1$
Conductor $507$
Sign $-0.971 - 0.236i$
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  + (0.600 + 0.799i)2-s + (−0.278 + 0.960i)4-s + (−0.464 + 0.885i)5-s + (−0.160 + 0.987i)7-s + (−0.935 + 0.354i)8-s + (−0.987 + 0.160i)10-s + (0.391 + 0.919i)11-s + (−0.885 + 0.464i)14-s + (−0.845 − 0.534i)16-s + (0.987 + 0.160i)17-s + (0.866 + 0.5i)19-s + (−0.721 − 0.692i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.568 − 0.822i)25-s + ⋯
L(s)  = 1  + (0.600 + 0.799i)2-s + (−0.278 + 0.960i)4-s + (−0.464 + 0.885i)5-s + (−0.160 + 0.987i)7-s + (−0.935 + 0.354i)8-s + (−0.987 + 0.160i)10-s + (0.391 + 0.919i)11-s + (−0.885 + 0.464i)14-s + (−0.845 − 0.534i)16-s + (0.987 + 0.160i)17-s + (0.866 + 0.5i)19-s + (−0.721 − 0.692i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.568 − 0.822i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.971 - 0.236i$
Analytic conductor: \(2.35449\)
Root analytic conductor: \(2.35449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 507,\ (0:\ ),\ -0.971 - 0.236i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1654066423 + 1.380110030i\)
\(L(\frac12)\) \(\approx\) \(-0.1654066423 + 1.380110030i\)
\(L(1)\) \(\approx\) \(0.7334185392 + 0.9565433260i\)
\(L(1)\) \(\approx\) \(0.7334185392 + 0.9565433260i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.600 + 0.799i)T \)
5 \( 1 + (-0.464 + 0.885i)T \)
7 \( 1 + (-0.160 + 0.987i)T \)
11 \( 1 + (0.391 + 0.919i)T \)
17 \( 1 + (0.987 + 0.160i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.799 + 0.600i)T \)
31 \( 1 + (-0.822 - 0.568i)T \)
37 \( 1 + (0.0804 - 0.996i)T \)
41 \( 1 + (-0.979 - 0.200i)T \)
43 \( 1 + (0.996 - 0.0804i)T \)
47 \( 1 + (-0.239 + 0.970i)T \)
53 \( 1 + (0.354 + 0.935i)T \)
59 \( 1 + (0.534 + 0.845i)T \)
61 \( 1 + (-0.632 - 0.774i)T \)
67 \( 1 + (0.960 - 0.278i)T \)
71 \( 1 + (0.316 + 0.948i)T \)
73 \( 1 + (0.992 + 0.120i)T \)
79 \( 1 + (-0.970 - 0.239i)T \)
83 \( 1 + (-0.663 - 0.748i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.999 + 0.0402i)T \)
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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

−23.16845630851112248093206296579, −22.308078973973269814493983681907, −21.36594445223078287399475047158, −20.58464192293843050696953171184, −19.88771298997663683256646575192, −19.33100528398940422291576045674, −18.31685413982691300852272739498, −17.06385877983122992910262696966, −16.33491048942351341160699562258, −15.425022107707978741795445556614, −14.2024881720289784675971211067, −13.58605694156595678130392242994, −12.83423019359130113229103000902, −11.746199024349266468764746542745, −11.29176079225151019401304423449, −10.058433010184990121801390735351, −9.3411694150090552871117415145, −8.209250584752794719184218906038, −7.10117157849371191269584576670, −5.75338330338533352743335326838, −4.970578485562869313400184058437, −3.799925730778230432740056568268, −3.3374568040626001632448992161, −1.54337110083706033807694244075, −0.638667752152692978546821552685, 2.187014011787763555613926671563, 3.26130642501370441924780992536, 4.11259733384722374033225427624, 5.4008967830729626779887003689, 6.15126253320208333821042156098, 7.20589989996755103760417796444, 7.82361847368463072368982335421, 8.99383747489666395346393270698, 9.94926538044213950320096656467, 11.317240128785939997957961954771, 12.213486418432322944168024677213, 12.6821538465881861225085945396, 14.23979959349991672553972463143, 14.57898551711967602787828528376, 15.43326288600410885431895721370, 16.14449152902980196830367931417, 17.12862897897523352547269236478, 18.34704262152536135833912539116, 18.53523127631857784647034301164, 19.92043693489988410181942229650, 20.90728684332272170189616547777, 21.95498323452170584897557860444, 22.48982478281943268918531128682, 23.063677744760698597741552683893, 24.074847289388984930208833962

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