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Properties

Label 1-4235-4235.1404-r1-0-0
Degree $1$
Conductor $4235$
Sign $-0.371 + 0.928i$
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  + (0.483 + 0.875i)2-s + (0.104 − 0.994i)3-s + (−0.532 + 0.846i)4-s + (0.921 − 0.389i)6-s + (−0.998 − 0.0570i)8-s + (−0.978 − 0.207i)9-s + (0.786 + 0.618i)12-s + (−0.985 + 0.170i)13-s + (−0.432 − 0.901i)16-s + (−0.595 − 0.803i)17-s + (−0.290 − 0.956i)18-s + (−0.00951 + 0.999i)19-s + (−0.723 + 0.690i)23-s + (−0.161 + 0.986i)24-s + (−0.625 − 0.780i)26-s + (−0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (0.483 + 0.875i)2-s + (0.104 − 0.994i)3-s + (−0.532 + 0.846i)4-s + (0.921 − 0.389i)6-s + (−0.998 − 0.0570i)8-s + (−0.978 − 0.207i)9-s + (0.786 + 0.618i)12-s + (−0.985 + 0.170i)13-s + (−0.432 − 0.901i)16-s + (−0.595 − 0.803i)17-s + (−0.290 − 0.956i)18-s + (−0.00951 + 0.999i)19-s + (−0.723 + 0.690i)23-s + (−0.161 + 0.986i)24-s + (−0.625 − 0.780i)26-s + (−0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.371 + 0.928i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ -0.371 + 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3264391419 + 0.4822110034i\)
\(L(\frac12)\) \(\approx\) \(0.3264391419 + 0.4822110034i\)
\(L(1)\) \(\approx\) \(0.9490564914 + 0.1252832086i\)
\(L(1)\) \(\approx\) \(0.9490564914 + 0.1252832086i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.483 + 0.875i)T \)
3 \( 1 + (0.104 - 0.994i)T \)
13 \( 1 + (-0.985 + 0.170i)T \)
17 \( 1 + (-0.595 - 0.803i)T \)
19 \( 1 + (-0.00951 + 0.999i)T \)
23 \( 1 + (-0.723 + 0.690i)T \)
29 \( 1 + (-0.198 - 0.980i)T \)
31 \( 1 + (-0.830 - 0.556i)T \)
37 \( 1 + (0.969 - 0.244i)T \)
41 \( 1 + (-0.0855 - 0.996i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (0.290 - 0.956i)T \)
53 \( 1 + (0.432 - 0.901i)T \)
59 \( 1 + (-0.905 + 0.424i)T \)
61 \( 1 + (0.999 - 0.0190i)T \)
67 \( 1 + (0.327 - 0.945i)T \)
71 \( 1 + (-0.736 - 0.676i)T \)
73 \( 1 + (-0.761 - 0.647i)T \)
79 \( 1 + (0.710 + 0.703i)T \)
83 \( 1 + (-0.0285 + 0.999i)T \)
89 \( 1 + (-0.995 - 0.0950i)T \)
97 \( 1 + (-0.774 - 0.633i)T \)
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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

−17.903341393963405674018422140863, −17.47698725344038773497019115692, −16.54592220664601947543654444720, −15.83064434492043340492699462830, −15.07400557323455836431821973389, −14.57920798937797676094153435061, −14.08789960173023066182902171930, −13.0404817837556770238395778224, −12.63178883701340084045475996080, −11.69231745183117564480549707524, −11.0810690291273134005382943674, −10.50921965080649562705385684884, −9.90247853919101438123738599770, −9.158639693443090970228517065187, −8.68939277322903705537004698204, −7.67681817741582743570412671476, −6.532097767173141182391221834554, −5.791244809544699314420370323861, −4.9779811504073662383949222626, −4.45117494590125933570863689826, −3.834470591158215871050929248527, −2.80443150181567911027316512748, −2.47575673850834649507323744972, −1.30382213888356692641769875789, −0.11298568081156677342765519866, 0.52660212249163493934035657974, 1.98116251608408501237980179096, 2.50423176786781853980381125648, 3.57162713215976279499869013203, 4.261368634696826485364724397773, 5.32314332920007359915956789941, 5.80126287925901908967348665053, 6.570764829452739071132099410922, 7.43607089481050504017351124257, 7.58971868396585452449554362313, 8.49833096278306918371795375902, 9.252088037706653724274606849738, 9.90799540009047018954510351735, 11.23406106665597527130659081510, 11.88036089329688020690117156419, 12.39791219565839350833266920958, 13.08941858381286224090210267504, 13.82120201453906557135157635828, 14.177923825966177344650216066222, 14.9884670513338968012212220853, 15.554439926589753435299287266971, 16.63791383474316249222233052987, 16.8598767073024986247423161791, 17.85231040319905826606838070548, 18.123461373512119054081630014470

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