Properties

Label 1-4235-4235.149-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: $\chi_{4235} (149, \cdot )$
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

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

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