Properties

Label 1-4235-4235.2253-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.918 + 0.394i$
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  + (0.791 − 0.610i)2-s + (−0.951 − 0.309i)3-s + (0.254 − 0.967i)4-s + (−0.941 + 0.336i)6-s + (−0.389 − 0.921i)8-s + (0.809 + 0.587i)9-s + (−0.540 + 0.841i)12-s + (−0.931 + 0.362i)13-s + (−0.870 − 0.491i)16-s + (−0.717 + 0.696i)17-s + (0.999 − 0.0285i)18-s + (0.897 + 0.441i)19-s + (−0.909 + 0.415i)23-s + (0.0855 + 0.996i)24-s + (−0.516 + 0.856i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.791 − 0.610i)2-s + (−0.951 − 0.309i)3-s + (0.254 − 0.967i)4-s + (−0.941 + 0.336i)6-s + (−0.389 − 0.921i)8-s + (0.809 + 0.587i)9-s + (−0.540 + 0.841i)12-s + (−0.931 + 0.362i)13-s + (−0.870 − 0.491i)16-s + (−0.717 + 0.696i)17-s + (0.999 − 0.0285i)18-s + (0.897 + 0.441i)19-s + (−0.909 + 0.415i)23-s + (0.0855 + 0.996i)24-s + (−0.516 + 0.856i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.918 + 0.394i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.918 + 0.394i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1610653308 - 0.7833255376i\)
\(L(\frac12)\) \(\approx\) \(-0.1610653308 - 0.7833255376i\)
\(L(1)\) \(\approx\) \(0.8588169811 - 0.5394910884i\)
\(L(1)\) \(\approx\) \(0.8588169811 - 0.5394910884i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.791 - 0.610i)T \)
3 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (-0.931 + 0.362i)T \)
17 \( 1 + (-0.717 + 0.696i)T \)
19 \( 1 + (0.897 + 0.441i)T \)
23 \( 1 + (-0.909 + 0.415i)T \)
29 \( 1 + (0.985 - 0.170i)T \)
31 \( 1 + (0.998 + 0.0570i)T \)
37 \( 1 + (0.633 - 0.774i)T \)
41 \( 1 + (0.564 - 0.825i)T \)
43 \( 1 + (0.755 - 0.654i)T \)
47 \( 1 + (-0.999 - 0.0285i)T \)
53 \( 1 + (-0.491 - 0.870i)T \)
59 \( 1 + (-0.564 - 0.825i)T \)
61 \( 1 + (-0.610 + 0.791i)T \)
67 \( 1 + (-0.281 + 0.959i)T \)
71 \( 1 + (-0.466 - 0.884i)T \)
73 \( 1 + (0.676 + 0.736i)T \)
79 \( 1 + (-0.974 + 0.226i)T \)
83 \( 1 + (0.980 - 0.198i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (-0.996 + 0.0855i)T \)
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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

−18.31279896132274633126803790385, −17.86920309918468404581224297457, −17.33730602762835236058806422836, −16.556447149347568835212411880570, −16.01375477783769315172779277580, −15.49335987148004204162488920728, −14.82160887307995295720992994553, −13.99597987129409408946519709894, −13.38163554714937304030828582028, −12.525810066761812606438441745924, −12.02609821774539312977941424537, −11.462451803583631173876721662485, −10.71329188594653743148136185773, −9.80018210923639361173000521078, −9.21271092771458790741989684254, −8.029950903862566474176421255857, −7.51468535431891950752983125408, −6.555874734043515748277351615347, −6.256742639403282987768250092657, −5.261619430823749728415094817411, −4.695548403649673019866272061437, −4.29826701024358601536191941966, −3.074489187698469107941832405536, −2.53259986310933797395517787857, −1.10618873871170031024023381085, 0.20106099976709779530603236485, 1.31509162080848956761149264533, 2.02989226217087380307174135227, 2.82305688097664788388644641314, 4.00302631175750338117575535214, 4.46712837247403860760466623921, 5.29363892237530013150280579154, 5.914239221907331300378907063003, 6.5781147158114351037485487659, 7.27915448386402747651122830761, 8.1274771471268078907734611979, 9.38165161983353233591025749746, 9.97722279163635390983171236863, 10.594700261566216492961276234787, 11.32998685522143419628819687848, 11.97932442520655362002937608118, 12.354525983089991822651366878417, 13.09134632365775011446522265750, 13.849691907493416961022565789275, 14.35779046570937108898206934462, 15.329232173141824612837794119841, 15.923416186725458643625430007505, 16.51210504560042401676117412885, 17.64346079423277798013808019554, 17.77994990659728918831882787767

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