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Properties

Label 1-4235-4235.698-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.914 + 0.404i$
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  + (0.999 − 0.00951i)2-s + (−0.207 + 0.978i)3-s + (0.999 − 0.0190i)4-s + (−0.198 + 0.980i)6-s + (0.999 − 0.0285i)8-s + (−0.913 − 0.406i)9-s + (−0.189 + 0.981i)12-s + (−0.996 − 0.0855i)13-s + (0.999 − 0.0380i)16-s + (0.836 + 0.548i)17-s + (−0.917 − 0.398i)18-s + (0.964 − 0.263i)19-s + (−0.618 − 0.786i)23-s + (−0.179 + 0.983i)24-s + (−0.997 − 0.0760i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.999 − 0.00951i)2-s + (−0.207 + 0.978i)3-s + (0.999 − 0.0190i)4-s + (−0.198 + 0.980i)6-s + (0.999 − 0.0285i)8-s + (−0.913 − 0.406i)9-s + (−0.189 + 0.981i)12-s + (−0.996 − 0.0855i)13-s + (0.999 − 0.0380i)16-s + (0.836 + 0.548i)17-s + (−0.917 − 0.398i)18-s + (0.964 − 0.263i)19-s + (−0.618 − 0.786i)23-s + (−0.179 + 0.983i)24-s + (−0.997 − 0.0760i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.914 + 0.404i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (698, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.914 + 0.404i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.237959488 + 0.6837236446i\)
\(L(\frac12)\) \(\approx\) \(3.237959488 + 0.6837236446i\)
\(L(1)\) \(\approx\) \(1.835369168 + 0.4099980646i\)
\(L(1)\) \(\approx\) \(1.835369168 + 0.4099980646i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.999 - 0.00951i)T \)
3 \( 1 + (-0.207 + 0.978i)T \)
13 \( 1 + (-0.996 - 0.0855i)T \)
17 \( 1 + (0.836 + 0.548i)T \)
19 \( 1 + (0.964 - 0.263i)T \)
23 \( 1 + (-0.618 - 0.786i)T \)
29 \( 1 + (-0.774 + 0.633i)T \)
31 \( 1 + (0.683 - 0.730i)T \)
37 \( 1 + (0.603 - 0.797i)T \)
41 \( 1 + (0.736 - 0.676i)T \)
43 \( 1 + (0.281 - 0.959i)T \)
47 \( 1 + (0.917 - 0.398i)T \)
53 \( 1 + (-0.0380 + 0.999i)T \)
59 \( 1 + (0.953 - 0.299i)T \)
61 \( 1 + (-0.861 - 0.508i)T \)
67 \( 1 + (-0.0950 + 0.995i)T \)
71 \( 1 + (-0.362 - 0.931i)T \)
73 \( 1 + (-0.768 - 0.640i)T \)
79 \( 1 + (0.991 + 0.132i)T \)
83 \( 1 + (-0.717 - 0.696i)T \)
89 \( 1 + (-0.888 - 0.458i)T \)
97 \( 1 + (0.336 + 0.941i)T \)
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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

−18.29164982115783852942300125386, −17.67445831907019633108353597117, −16.813899093501783843564582610906, −16.38357147044821358632555810148, −15.51823260706960481642328847704, −14.68922956007399302589689789250, −14.09487582062630734340288500306, −13.65572786962497002608022440928, −12.82648709998832501731380835339, −12.28815555519563290160771058300, −11.62802173158860644037621308814, −11.309736763232835782836299779034, −10.097991084405377111410416209424, −9.58413363718216152920549495885, −8.2513155076358171623557451075, −7.53522423445744035333629959810, −7.2704577796306137491134165551, −6.28150587621278986289682229064, −5.673219447520369008279292530607, −5.08117467457002995296658358573, −4.222464892628391316491512789328, −3.1124281377145584642515664078, −2.67335232927029126661364569099, −1.70154806127200092760490037779, −0.955849773056232975045170054111, 0.75006073034508735695304080645, 2.12386525460642528504486631452, 2.80955616165694974853124600698, 3.63951380906114152725248658740, 4.23552602158522255114970171248, 4.97831834522379478336160661752, 5.64927529262789949770739934583, 6.12563255890854954483855182481, 7.27298103348577700352793907983, 7.759095888045051328174842790556, 8.86826937415220818951122550022, 9.67765946673691396954803994102, 10.35101289530213139386749524008, 10.84765086515291806047943784068, 11.79864060303603728414201932380, 12.1551074019586387596018384503, 12.910550659332212471493062762517, 13.91710753573372727623724596528, 14.41794147505805478600623270431, 14.95145763892251163650148416402, 15.641142870106199560313251340850, 16.24500492925122314479138607921, 16.88743628773228475790921810010, 17.379158418225431460812270715677, 18.44725838205704206807053160940

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