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Properties

Label 1-4235-4235.1437-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.523 - 0.851i$
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  + (0.768 − 0.640i)2-s + (−0.406 − 0.913i)3-s + (0.179 − 0.983i)4-s + (−0.897 − 0.441i)6-s + (−0.491 − 0.870i)8-s + (−0.669 + 0.743i)9-s + (−0.971 + 0.235i)12-s + (0.999 − 0.0285i)13-s + (−0.935 − 0.353i)16-s + (0.0190 + 0.999i)17-s + (−0.0380 + 0.999i)18-s + (0.820 − 0.572i)19-s + (−0.998 − 0.0475i)23-s + (−0.595 + 0.803i)24-s + (0.749 − 0.662i)26-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (0.768 − 0.640i)2-s + (−0.406 − 0.913i)3-s + (0.179 − 0.983i)4-s + (−0.897 − 0.441i)6-s + (−0.491 − 0.870i)8-s + (−0.669 + 0.743i)9-s + (−0.971 + 0.235i)12-s + (0.999 − 0.0285i)13-s + (−0.935 − 0.353i)16-s + (0.0190 + 0.999i)17-s + (−0.0380 + 0.999i)18-s + (0.820 − 0.572i)19-s + (−0.998 − 0.0475i)23-s + (−0.595 + 0.803i)24-s + (0.749 − 0.662i)26-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.523 - 0.851i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.523 - 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.217972644 - 2.179017109i\)
\(L(\frac12)\) \(\approx\) \(1.217972644 - 2.179017109i\)
\(L(1)\) \(\approx\) \(1.127537092 - 0.9632178341i\)
\(L(1)\) \(\approx\) \(1.127537092 - 0.9632178341i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.768 - 0.640i)T \)
3 \( 1 + (-0.406 - 0.913i)T \)
13 \( 1 + (0.999 - 0.0285i)T \)
17 \( 1 + (0.0190 + 0.999i)T \)
19 \( 1 + (0.820 - 0.572i)T \)
23 \( 1 + (-0.998 - 0.0475i)T \)
29 \( 1 + (0.974 + 0.226i)T \)
31 \( 1 + (0.997 - 0.0760i)T \)
37 \( 1 + (0.132 - 0.991i)T \)
41 \( 1 + (-0.696 + 0.717i)T \)
43 \( 1 + (0.909 + 0.415i)T \)
47 \( 1 + (0.0380 + 0.999i)T \)
53 \( 1 + (0.353 + 0.935i)T \)
59 \( 1 + (0.969 - 0.244i)T \)
61 \( 1 + (-0.345 - 0.938i)T \)
67 \( 1 + (0.618 - 0.786i)T \)
71 \( 1 + (-0.921 + 0.389i)T \)
73 \( 1 + (0.893 + 0.449i)T \)
79 \( 1 + (0.953 + 0.299i)T \)
83 \( 1 + (0.967 - 0.254i)T \)
89 \( 1 + (-0.981 - 0.189i)T \)
97 \( 1 + (-0.113 + 0.993i)T \)
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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

−18.05533131376649788040273236860, −17.99338511980140293247866353999, −16.94650020431971650188354709719, −16.36373533790111066893882603754, −15.8406785667215819319684739971, −15.43503171827007801215348739645, −14.58914148769450362006217614657, −13.75639171386748376507925172997, −13.61726670793349760304647593430, −12.27099226001553031355579013240, −11.85524067091132303136024489965, −11.31829222830390294276809905150, −10.31654889926824928599494770965, −9.76044641317528387636250495637, −8.71884826062211386487224601106, −8.30599037761209676362638494704, −7.29872357777116161267385680109, −6.48644221251325908359854066322, −5.8776203023543231434582047319, −5.21759922056201510028692082671, −4.54510755548495654411174431083, −3.77118851998727384995754642631, −3.23878705793525180458176865770, −2.308230389105997667618924699560, −0.833368786821584569497221288791, 0.769058901977073060526885511008, 1.390271298764148699720797353491, 2.25535972395292706809503902167, 3.009945622983113100111389415232, 3.88332790166418151391109684439, 4.70365604743707218369885434924, 5.53709423887492262276321587317, 6.22248208802218320947283676150, 6.58656716600596884635978633576, 7.662318130086953237448988972437, 8.34785067617871329652509814432, 9.25117533326906334068192599186, 10.20147686949149524584770961863, 10.84480332719314174201525613033, 11.42211066906604269949469523202, 12.09700115349189504543636525792, 12.67878288136314240620551471923, 13.29295925231267151866831850211, 13.97287668657566298425484154027, 14.33736533273721295313545893279, 15.509021564446966705146536573060, 15.914334605289971877120474896823, 16.822878096570783329878421097050, 17.85229556523434930822558803848, 18.079624363980754370782660762435

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