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Properties

Label 1-4235-4235.383-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.943 + 0.330i$
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  + (0.780 + 0.625i)2-s + (0.743 − 0.669i)3-s + (0.217 + 0.976i)4-s + (0.998 − 0.0570i)6-s + (−0.441 + 0.897i)8-s + (0.104 − 0.994i)9-s + (0.814 + 0.580i)12-s + (−0.980 − 0.198i)13-s + (−0.905 + 0.424i)16-s + (−0.924 + 0.380i)17-s + (0.703 − 0.710i)18-s + (0.997 + 0.0760i)19-s + (−0.189 + 0.981i)23-s + (0.272 + 0.962i)24-s + (−0.640 − 0.768i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.780 + 0.625i)2-s + (0.743 − 0.669i)3-s + (0.217 + 0.976i)4-s + (0.998 − 0.0570i)6-s + (−0.441 + 0.897i)8-s + (0.104 − 0.994i)9-s + (0.814 + 0.580i)12-s + (−0.980 − 0.198i)13-s + (−0.905 + 0.424i)16-s + (−0.924 + 0.380i)17-s + (0.703 − 0.710i)18-s + (0.997 + 0.0760i)19-s + (−0.189 + 0.981i)23-s + (0.272 + 0.962i)24-s + (−0.640 − 0.768i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.943 + 0.330i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.943 + 0.330i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2593211865 + 1.525604356i\)
\(L(\frac12)\) \(\approx\) \(0.2593211865 + 1.525604356i\)
\(L(1)\) \(\approx\) \(1.477834219 + 0.5322638439i\)
\(L(1)\) \(\approx\) \(1.477834219 + 0.5322638439i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.780 + 0.625i)T \)
3 \( 1 + (0.743 - 0.669i)T \)
13 \( 1 + (-0.980 - 0.198i)T \)
17 \( 1 + (-0.924 + 0.380i)T \)
19 \( 1 + (0.997 + 0.0760i)T \)
23 \( 1 + (-0.189 + 0.981i)T \)
29 \( 1 + (0.0285 + 0.999i)T \)
31 \( 1 + (-0.00951 + 0.999i)T \)
37 \( 1 + (-0.917 + 0.398i)T \)
41 \( 1 + (-0.774 + 0.633i)T \)
43 \( 1 + (-0.989 - 0.142i)T \)
47 \( 1 + (-0.703 - 0.710i)T \)
53 \( 1 + (0.424 - 0.905i)T \)
59 \( 1 + (-0.935 + 0.353i)T \)
61 \( 1 + (-0.988 + 0.151i)T \)
67 \( 1 + (0.458 + 0.888i)T \)
71 \( 1 + (0.941 - 0.336i)T \)
73 \( 1 + (-0.603 - 0.797i)T \)
79 \( 1 + (-0.999 + 0.0380i)T \)
83 \( 1 + (0.226 - 0.974i)T \)
89 \( 1 + (0.723 + 0.690i)T \)
97 \( 1 + (0.717 + 0.696i)T \)
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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

−18.44375119372923189352372137427, −17.30018475913765675757135631828, −16.54806260874424639708404749983, −15.64334212042253415161604408732, −15.363729072969478115434097401313, −14.550695037903292024685082236689, −13.956781240953870734217453523448, −13.508471992811757293521059961714, −12.67688298213801603856299542442, −11.914190587361608086990247996, −11.25863561888160376301973988436, −10.523823482604880681823069325680, −9.79273424600444292161006605745, −9.40365262591986906652989302341, −8.55593711320080042358194793315, −7.590440223722932128491745832621, −6.86691727050591704004845030270, −5.92382474285178005547159088081, −4.97897669420969413854661218364, −4.5915988806247851187866443707, −3.82924681373716737094127555565, −2.98790846718710450101676843713, −2.39304691430315259406493350910, −1.71428776200393011956434141213, −0.24235753934447917440499415552, 1.48444379937464428562287485435, 2.18335243731420415616433232687, 3.28003473415597452150859929250, 3.419617082119777785306845296679, 4.70112582781374528127603818296, 5.22544583910932629718709785562, 6.20133727426832303381080691427, 6.97887362959101051072344986050, 7.30848759277821943151397936683, 8.17677227315849366635904381070, 8.72746415784100972594795050278, 9.51369020741495580710949787111, 10.412924563105079646589943633278, 11.614176244831013956113579758626, 12.002961107280558458697030331287, 12.75508238208687919699036516915, 13.39840070398548577212273035202, 13.86937092660658330243570776779, 14.61535593601807702911528989715, 15.13409714497323867830344808606, 15.74325610114885979403794263878, 16.54473564786673532020255133671, 17.40856195014457074287304141696, 17.87333002272667982984002435332, 18.516150107564356290963245324152

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