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Properties

Label 1-4235-4235.103-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.127 - 0.991i$
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  + (−0.986 − 0.161i)2-s + (−0.406 − 0.913i)3-s + (0.948 + 0.318i)4-s + (0.254 + 0.967i)6-s + (−0.884 − 0.466i)8-s + (−0.669 + 0.743i)9-s + (−0.0950 − 0.995i)12-s + (0.113 − 0.993i)13-s + (0.797 + 0.603i)16-s + (0.901 − 0.432i)17-s + (0.780 − 0.625i)18-s + (−0.179 − 0.983i)19-s + (−0.945 − 0.327i)23-s + (−0.0665 + 0.997i)24-s + (−0.272 + 0.962i)26-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.986 − 0.161i)2-s + (−0.406 − 0.913i)3-s + (0.948 + 0.318i)4-s + (0.254 + 0.967i)6-s + (−0.884 − 0.466i)8-s + (−0.669 + 0.743i)9-s + (−0.0950 − 0.995i)12-s + (0.113 − 0.993i)13-s + (0.797 + 0.603i)16-s + (0.901 − 0.432i)17-s + (0.780 − 0.625i)18-s + (−0.179 − 0.983i)19-s + (−0.945 − 0.327i)23-s + (−0.0665 + 0.997i)24-s + (−0.272 + 0.962i)26-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.127 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.127 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.127 - 0.991i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.127 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6759484861 - 0.5945221336i\)
\(L(\frac12)\) \(\approx\) \(0.6759484861 - 0.5945221336i\)
\(L(1)\) \(\approx\) \(0.5894910018 - 0.2562134488i\)
\(L(1)\) \(\approx\) \(0.5894910018 - 0.2562134488i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.986 - 0.161i)T \)
3 \( 1 + (-0.406 - 0.913i)T \)
13 \( 1 + (0.113 - 0.993i)T \)
17 \( 1 + (0.901 - 0.432i)T \)
19 \( 1 + (-0.179 - 0.983i)T \)
23 \( 1 + (-0.945 - 0.327i)T \)
29 \( 1 + (-0.610 + 0.791i)T \)
31 \( 1 + (0.217 + 0.976i)T \)
37 \( 1 + (0.999 - 0.00951i)T \)
41 \( 1 + (0.998 + 0.0570i)T \)
43 \( 1 + (0.989 - 0.142i)T \)
47 \( 1 + (-0.780 - 0.625i)T \)
53 \( 1 + (0.603 + 0.797i)T \)
59 \( 1 + (0.449 - 0.893i)T \)
61 \( 1 + (0.935 + 0.353i)T \)
67 \( 1 + (0.998 - 0.0475i)T \)
71 \( 1 + (-0.0285 + 0.999i)T \)
73 \( 1 + (0.730 + 0.683i)T \)
79 \( 1 + (-0.640 - 0.768i)T \)
83 \( 1 + (0.856 - 0.516i)T \)
89 \( 1 + (0.235 + 0.971i)T \)
97 \( 1 + (0.441 + 0.897i)T \)
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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

−18.42302942292091775702471777107, −17.80012666677763015687655615338, −16.9428640998244701764866080782, −16.646245866047942939919315634472, −16.03598850348857771461055798713, −15.3795995123820427168022945614, −14.526544740664730583393201364819, −14.273641597292047404095929810508, −12.92195407916988857243251440196, −11.974831565378414718424143748303, −11.56464527340264759249029932907, −10.887419313387748038398712042396, −10.07169031365091236921791959523, −9.65722350161162163659841981919, −9.087391550745660205884040714707, −8.08881979230957026550672146781, −7.70092912476811389439755967115, −6.4921088894690608941819529708, −6.00015749565959477674721930989, −5.41398961523085331903436857181, −4.18041622777003699195695351578, −3.73692471489600385486729830892, −2.58443902899648166734259567625, −1.75278659157081021652500434441, −0.68744690678350394513442113044, 0.616823729516395882898695344990, 1.156772791987928007831891125798, 2.268760633480246910622348889948, 2.783642502295861040855169260086, 3.7313252534091529045588124382, 5.102166636536550008023688613418, 5.74573962640245733891452045897, 6.51962844969869972034230871601, 7.20713574649080000507621072907, 7.82041733698918417836706498037, 8.38129220748493268989229216745, 9.181128150115190101684524078150, 10.03794625821371316826263984402, 10.71796634261700777849630318928, 11.298639199368856634050264708352, 12.01814711227782965832021041481, 12.65599457484677143342542950751, 13.1507945668963144442555460729, 14.20154528063836742359623417412, 14.83502108868717309889909412750, 15.9292065900686842549811992259, 16.23750938439858852337271371928, 17.14452886625290509025920231515, 17.67630816197399317308471906759, 18.17562598851070749205582060014

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