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Properties

Label 1-4235-4235.507-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.234 - 0.972i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (0.189 + 0.981i)2-s + (0.866 − 0.5i)3-s + (−0.928 + 0.371i)4-s + (0.654 + 0.755i)6-s + (−0.540 − 0.841i)8-s + (0.5 − 0.866i)9-s + (−0.618 + 0.786i)12-s + (−0.989 − 0.142i)13-s + (0.723 − 0.690i)16-s + (−0.814 − 0.580i)17-s + (0.945 + 0.327i)18-s + (0.580 + 0.814i)19-s + (−0.690 − 0.723i)23-s + (−0.888 − 0.458i)24-s + (−0.0475 − 0.998i)26-s − i·27-s + ⋯
L(s)  = 1  + (0.189 + 0.981i)2-s + (0.866 − 0.5i)3-s + (−0.928 + 0.371i)4-s + (0.654 + 0.755i)6-s + (−0.540 − 0.841i)8-s + (0.5 − 0.866i)9-s + (−0.618 + 0.786i)12-s + (−0.989 − 0.142i)13-s + (0.723 − 0.690i)16-s + (−0.814 − 0.580i)17-s + (0.945 + 0.327i)18-s + (0.580 + 0.814i)19-s + (−0.690 − 0.723i)23-s + (−0.888 − 0.458i)24-s + (−0.0475 − 0.998i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.234 - 0.972i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.234 - 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4424455116 - 0.5620687418i\)
\(L(\frac12)\) \(\approx\) \(0.4424455116 - 0.5620687418i\)
\(L(1)\) \(\approx\) \(1.074815495 + 0.2321389583i\)
\(L(1)\) \(\approx\) \(1.074815495 + 0.2321389583i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.189 + 0.981i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.989 - 0.142i)T \)
17 \( 1 + (-0.814 - 0.580i)T \)
19 \( 1 + (0.580 + 0.814i)T \)
23 \( 1 + (-0.690 - 0.723i)T \)
29 \( 1 + (-0.415 + 0.909i)T \)
31 \( 1 + (0.786 - 0.618i)T \)
37 \( 1 + (-0.371 + 0.928i)T \)
41 \( 1 + (0.654 + 0.755i)T \)
43 \( 1 + (-0.540 - 0.841i)T \)
47 \( 1 + (-0.945 + 0.327i)T \)
53 \( 1 + (0.690 - 0.723i)T \)
59 \( 1 + (0.981 + 0.189i)T \)
61 \( 1 + (0.327 + 0.945i)T \)
67 \( 1 + (-0.945 - 0.327i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (-0.971 + 0.235i)T \)
79 \( 1 + (0.888 - 0.458i)T \)
83 \( 1 + (-0.281 - 0.959i)T \)
89 \( 1 + (-0.995 - 0.0950i)T \)
97 \( 1 + (-0.540 - 0.841i)T \)
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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

−18.955203010127902013583806703970, −17.81940935152766243619711270744, −17.558519496232344468099983495636, −16.49593538949270310042274970747, −15.59473589527337909546398428154, −15.10127255311732174754912517686, −14.34816125393808276622150729308, −13.769992027908061622924292642969, −13.20129461085624573839576046118, −12.48799218850355950585962108745, −11.65432236471936980195517801621, −11.037388559239917867121756442605, −10.216759555198092533794520956243, −9.68304223627355748513106595368, −9.13629009090755056415993101521, −8.39135682168271611808065083028, −7.68803314964213847946971896101, −6.773638803737305796721186988594, −5.58111557570717707061938135746, −4.91457085515101699038702928813, −4.17600041133955340846282440665, −3.634044899221589020907985775387, −2.619194036108559836507754735847, −2.26452557969238653311631019336, −1.30019542749778980761206831296, 0.15627141018417155514317160146, 1.39450850956477432597148623966, 2.47781525530626103868750871098, 3.16384468904545356393486089354, 4.06661739099334437849635922280, 4.741663821274916148199799352082, 5.602807362369120224002132453898, 6.519577787085731492901544740354, 7.02302348090694359928295228184, 7.74506241068439415443210150536, 8.29249370511436238778068147404, 8.9887376408717403379801192585, 9.73464033055091698924989282232, 10.233571791029126281843607526950, 11.724867946850130465694371243697, 12.203255305274905547419223564106, 13.02462443008139349267968610668, 13.53314663146819736835074584751, 14.22741770439567926559165965840, 14.78669322555677314760717339574, 15.26106067372987642631084283333, 16.17477974258309434190237899200, 16.63014448463656334396947846539, 17.68534510127646844325010986491, 18.025308550135552594496164198723

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