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Properties

Label 1-4235-4235.739-r1-0-0
Degree $1$
Conductor $4235$
Sign $0.112 + 0.993i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (0.00951 − 0.999i)2-s + (0.978 − 0.207i)3-s + (−0.999 − 0.0190i)4-s + (−0.198 − 0.980i)6-s + (−0.0285 + 0.999i)8-s + (0.913 − 0.406i)9-s + (−0.981 + 0.189i)12-s + (0.0855 + 0.996i)13-s + (0.999 + 0.0380i)16-s + (0.548 + 0.836i)17-s + (−0.398 − 0.917i)18-s + (−0.964 − 0.263i)19-s + (0.786 + 0.618i)23-s + (0.179 + 0.983i)24-s + (0.997 − 0.0760i)26-s + (0.809 − 0.587i)27-s + ⋯
L(s)  = 1  + (0.00951 − 0.999i)2-s + (0.978 − 0.207i)3-s + (−0.999 − 0.0190i)4-s + (−0.198 − 0.980i)6-s + (−0.0285 + 0.999i)8-s + (0.913 − 0.406i)9-s + (−0.981 + 0.189i)12-s + (0.0855 + 0.996i)13-s + (0.999 + 0.0380i)16-s + (0.548 + 0.836i)17-s + (−0.398 − 0.917i)18-s + (−0.964 − 0.263i)19-s + (0.786 + 0.618i)23-s + (0.179 + 0.983i)24-s + (0.997 − 0.0760i)26-s + (0.809 − 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.112 + 0.993i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ 0.112 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6795555529 + 0.6069786514i\)
\(L(\frac12)\) \(\approx\) \(0.6795555529 + 0.6069786514i\)
\(L(1)\) \(\approx\) \(1.108847392 - 0.4902678263i\)
\(L(1)\) \(\approx\) \(1.108847392 - 0.4902678263i\)

Euler product

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

−18.085896817819966140834224445894, −17.25112624299624342542727691180, −16.54236700184302840323894603106, −15.8951614791844842799457518436, −15.346034096975427359001493897926, −14.57417849145570688225629359880, −14.3197282661824557147617155250, −13.44953012535552839819664966606, −12.75941887494247186499009415336, −12.383192270306448175831539418939, −10.82165312956105620826605824993, −10.40358608674350235889110375989, −9.42702615776841103889250380030, −8.98658235603637889843830435849, −8.277629047115477957396758210319, −7.620246243737799933896012606, −7.073124522695934735692278270069, −6.22470903122618971865565257808, −5.210057737031448625303636022830, −4.81732005510963206913008515742, −3.682510430119926202606718999941, −3.3064966309733679111520724405, −2.23825390097839601849356484391, −1.12956342503302185840211139567, −0.11428779833301585143955567982, 1.12731462761079396838350922602, 1.814254501427457003575096600737, 2.4053101915744283347473441557, 3.36544667984838001517566637404, 3.91987058684596648595341354828, 4.56252510951074510485964482038, 5.575221112981663086070032637640, 6.55189396516232538046417859340, 7.3998598485800771524239564523, 8.26053350256576428334207220446, 8.721761564022629106860109715959, 9.53167255650008301276149145719, 9.91730348247325386662723518413, 10.92248547365206891794960885126, 11.45699099265462865230688549006, 12.37988626275512949363962793032, 12.91579704418972088033678466621, 13.51895246773454122685810858149, 14.10571223627717538399926087453, 14.97298826177170735857063871105, 15.18102193350934914917536789286, 16.557980205823111402570848499189, 17.07161753090087134717070243707, 17.939407717118667884104071884833, 18.7409980444978064327621702899

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