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Properties

Label 1-4235-4235.824-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.867 - 0.497i$
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  + (−0.327 + 0.945i)2-s + (−0.5 + 0.866i)3-s + (−0.786 − 0.618i)4-s + (−0.654 − 0.755i)6-s + (0.841 − 0.540i)8-s + (−0.5 − 0.866i)9-s + (0.928 − 0.371i)12-s + (0.142 − 0.989i)13-s + (0.235 + 0.971i)16-s + (0.995 + 0.0950i)17-s + (0.981 − 0.189i)18-s + (−0.995 + 0.0950i)19-s + (−0.235 − 0.971i)23-s + (0.0475 + 0.998i)24-s + (0.888 + 0.458i)26-s + 27-s + ⋯
L(s)  = 1  + (−0.327 + 0.945i)2-s + (−0.5 + 0.866i)3-s + (−0.786 − 0.618i)4-s + (−0.654 − 0.755i)6-s + (0.841 − 0.540i)8-s + (−0.5 − 0.866i)9-s + (0.928 − 0.371i)12-s + (0.142 − 0.989i)13-s + (0.235 + 0.971i)16-s + (0.995 + 0.0950i)17-s + (0.981 − 0.189i)18-s + (−0.995 + 0.0950i)19-s + (−0.235 − 0.971i)23-s + (0.0475 + 0.998i)24-s + (0.888 + 0.458i)26-s + 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.867 - 0.497i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.867 - 0.497i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5278653887 - 0.1405737260i\)
\(L(\frac12)\) \(\approx\) \(0.5278653887 - 0.1405737260i\)
\(L(1)\) \(\approx\) \(0.5720518640 + 0.3028335746i\)
\(L(1)\) \(\approx\) \(0.5720518640 + 0.3028335746i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.327 + 0.945i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.142 - 0.989i)T \)
17 \( 1 + (0.995 + 0.0950i)T \)
19 \( 1 + (-0.995 + 0.0950i)T \)
23 \( 1 + (-0.235 - 0.971i)T \)
29 \( 1 + (-0.415 + 0.909i)T \)
31 \( 1 + (-0.928 - 0.371i)T \)
37 \( 1 + (0.786 - 0.618i)T \)
41 \( 1 + (-0.654 - 0.755i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (0.981 + 0.189i)T \)
53 \( 1 + (-0.235 + 0.971i)T \)
59 \( 1 + (0.327 + 0.945i)T \)
61 \( 1 + (0.981 + 0.189i)T \)
67 \( 1 + (-0.981 + 0.189i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (-0.723 + 0.690i)T \)
79 \( 1 + (-0.0475 + 0.998i)T \)
83 \( 1 + (0.959 - 0.281i)T \)
89 \( 1 + (-0.580 + 0.814i)T \)
97 \( 1 + (0.841 - 0.540i)T \)
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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

−18.624690189158669207735950746225, −17.854580189069506875920339390906, −17.23990022667405457872314732820, −16.684792940057709305477213486275, −16.08107352779057040646448434083, −14.7636049706134322815188122366, −14.14992599940663035796137469582, −13.36720431529448848829899766504, −12.93868055205580724041454777753, −12.146387005761380297878521607700, −11.57438132309775864530889503221, −11.14600119429165759505093981736, −10.27708281394562168331014750149, −9.56605667460940212366016202376, −8.80734216740305991859124913310, −7.9944390994065723243077673066, −7.4833200662088813567137057888, −6.592000791056708798914886275473, −5.78419784116448884753512155437, −4.96627899889315326139603880024, −4.13118143849258165832926296782, −3.30913659665212616582084511385, −2.30412643692882737619861702486, −1.73700213564778529770507603979, −0.948772216335265318458597260234, 0.23469745177706486236864272224, 1.17525174123586521598141535168, 2.55979252083766441425686212313, 3.726365592258501403082919492702, 4.18014140177009303245542964996, 5.18378987673211255235950078339, 5.67178183292409805127260630376, 6.22569655084422629274067657779, 7.18134297607174832674449827320, 7.88337074354560137843025594915, 8.76468956071936738039927070454, 9.16138036645917269818469062461, 10.25461551655041575251047868746, 10.42641993031731677474601750925, 11.17034691537397299174651803897, 12.41040633766129778081995878294, 12.73904637544962769073192726985, 13.852505532932393225027875750598, 14.645671569376452824711833873667, 14.95925310013620248609188468013, 15.69202277511988811181604603540, 16.3984556858145944479791940334, 16.81817980165843920600908583960, 17.436029168485304700072517633257, 18.17617592712086235491480384510

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