Properties

Label 1-4235-4235.1014-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.989 + 0.147i$
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  + (0.610 + 0.791i)2-s + (0.309 − 0.951i)3-s + (−0.254 + 0.967i)4-s + (0.941 − 0.336i)6-s + (−0.921 + 0.389i)8-s + (−0.809 − 0.587i)9-s + (0.841 + 0.540i)12-s + (0.362 + 0.931i)13-s + (−0.870 − 0.491i)16-s + (−0.696 − 0.717i)17-s + (−0.0285 − 0.999i)18-s + (0.897 + 0.441i)19-s + (−0.415 − 0.909i)23-s + (0.0855 + 0.996i)24-s + (−0.516 + 0.856i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (0.610 + 0.791i)2-s + (0.309 − 0.951i)3-s + (−0.254 + 0.967i)4-s + (0.941 − 0.336i)6-s + (−0.921 + 0.389i)8-s + (−0.809 − 0.587i)9-s + (0.841 + 0.540i)12-s + (0.362 + 0.931i)13-s + (−0.870 − 0.491i)16-s + (−0.696 − 0.717i)17-s + (−0.0285 − 0.999i)18-s + (0.897 + 0.441i)19-s + (−0.415 − 0.909i)23-s + (0.0855 + 0.996i)24-s + (−0.516 + 0.856i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.989 + 0.147i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1014, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.989 + 0.147i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.366749376 + 0.1754507780i\)
\(L(\frac12)\) \(\approx\) \(2.366749376 + 0.1754507780i\)
\(L(1)\) \(\approx\) \(1.454692712 + 0.2416920079i\)
\(L(1)\) \(\approx\) \(1.454692712 + 0.2416920079i\)

Euler product

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

−18.45317427067698069650432755739, −17.686467947541027871428056814102, −17.12161330947806235205388872018, −15.79820612580953515994246668489, −15.66476595649738573404723602221, −15.01767475335982038034427130795, −14.169249334425897786209110853335, −13.59306863409079893621827739594, −13.092953758167358896453448756615, −12.06405528335018243946757654521, −11.483725138950258647095769226761, −10.82725828402732028098109469653, −9.9664070642228689908743836078, −9.91451748508773284128391574898, −8.610089569119615111291810271206, −8.43929253187854454970672390708, −7.10846736593164856824699274097, −6.13447485433892339441792706584, −5.39841406602024436645118573598, −4.87226792167261732292901128183, −4.00831274542036513405435518369, −3.39248828018231696527883534783, −2.78254552444327750459520926366, −1.89404736066740192234210815186, −0.80923093948501018607648785198, 0.64998536148852936210302273960, 1.87528284852974495449682218381, 2.652319783321132597523185620534, 3.41293976148318023934546800877, 4.29161060522257179050123302254, 5.05437835122121773845205731965, 5.94939440425382836718788586006, 6.66977585673824294766611572590, 6.97554963728970191064531120177, 7.90823187860662304844044225458, 8.49154395117215721408611224956, 9.06691239642129177791376199754, 9.967906670774867703777938810551, 11.224674716576027069626059566, 11.92549085680664890260765033302, 12.23419587362208528491331341736, 13.23323378984668498254742375427, 13.72925277445511856759630942409, 14.16644815970456797173885002202, 14.8112666469287700833433863104, 15.77623013736898071078405812332, 16.202678854418627044574415691792, 17.02815751527999575477938603114, 17.745611133011216691459620193662, 18.26788647653816733023415738936

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