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Properties

Label 1-4235-4235.38-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.827 - 0.560i$
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  + (0.572 − 0.820i)2-s + (−0.743 − 0.669i)3-s + (−0.345 − 0.938i)4-s + (−0.974 + 0.226i)6-s + (−0.967 − 0.254i)8-s + (0.104 + 0.994i)9-s + (−0.371 + 0.928i)12-s + (0.717 − 0.696i)13-s + (−0.761 + 0.647i)16-s + (−0.508 + 0.861i)17-s + (0.875 + 0.483i)18-s + (−0.217 − 0.976i)19-s + (0.971 + 0.235i)23-s + (0.548 + 0.836i)24-s + (−0.161 − 0.986i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.572 − 0.820i)2-s + (−0.743 − 0.669i)3-s + (−0.345 − 0.938i)4-s + (−0.974 + 0.226i)6-s + (−0.967 − 0.254i)8-s + (0.104 + 0.994i)9-s + (−0.371 + 0.928i)12-s + (0.717 − 0.696i)13-s + (−0.761 + 0.647i)16-s + (−0.508 + 0.861i)17-s + (0.875 + 0.483i)18-s + (−0.217 − 0.976i)19-s + (0.971 + 0.235i)23-s + (0.548 + 0.836i)24-s + (−0.161 − 0.986i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.827 - 0.560i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.827 - 0.560i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4667617997 - 1.521636712i\)
\(L(\frac12)\) \(\approx\) \(0.4667617997 - 1.521636712i\)
\(L(1)\) \(\approx\) \(0.7880684785 - 0.7389320414i\)
\(L(1)\) \(\approx\) \(0.7880684785 - 0.7389320414i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.572 - 0.820i)T \)
3 \( 1 + (-0.743 - 0.669i)T \)
13 \( 1 + (0.717 - 0.696i)T \)
17 \( 1 + (-0.508 + 0.861i)T \)
19 \( 1 + (-0.217 - 0.976i)T \)
23 \( 1 + (0.971 + 0.235i)T \)
29 \( 1 + (-0.993 + 0.113i)T \)
31 \( 1 + (0.532 - 0.846i)T \)
37 \( 1 + (0.556 + 0.830i)T \)
41 \( 1 + (0.921 + 0.389i)T \)
43 \( 1 + (-0.540 + 0.841i)T \)
47 \( 1 + (-0.875 + 0.483i)T \)
53 \( 1 + (0.647 - 0.761i)T \)
59 \( 1 + (0.797 + 0.603i)T \)
61 \( 1 + (0.905 + 0.424i)T \)
67 \( 1 + (0.189 - 0.981i)T \)
71 \( 1 + (0.198 - 0.980i)T \)
73 \( 1 + (-0.475 - 0.879i)T \)
79 \( 1 + (0.625 + 0.780i)T \)
83 \( 1 + (0.791 - 0.610i)T \)
89 \( 1 + (0.580 + 0.814i)T \)
97 \( 1 + (-0.0570 - 0.998i)T \)
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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

−18.39024521662572655234191728645, −17.77452053903402149846674209433, −17.072510074258606985205293568455, −16.44008344367716580328373282467, −16.04119358026843975197418482105, −15.392407834121268935683095950565, −14.6246372471582222387657519393, −14.12648979067347206660138554643, −13.19409246431258309615912864623, −12.63926658200965729406038837773, −11.73737267511514128349544618580, −11.34305877426515117008126480707, −10.501488428280878955831872542172, −9.54099534398890477239325933975, −8.959261694513898559975044854107, −8.295447666122251035594201169692, −7.1293257461977212770830009845, −6.76353938818165186624184289442, −5.88582762536006539552470818443, −5.364682599017941902876931951852, −4.57364395056595164597315499153, −3.94467449617585457004940231435, −3.319075724122937997166620970124, −2.19075355008352528729319804243, −0.789820568230095305523155659239, 0.5720606002749039150191008701, 1.30813829374736472085177369461, 2.16971074276773192446343260190, 2.93959350869879577318435036963, 3.86630046492044567775580274382, 4.73286310022880854047389682222, 5.30819487481565997733151097586, 6.22155543818745845682539352128, 6.50448602820854035710746113636, 7.621641047542396077148394787222, 8.41462584884034935545723522845, 9.26372493083172655439662896867, 10.12105900068514399947392066108, 10.94810573392553401670637070680, 11.1974141739611149724463564206, 11.884951489077827856820468362736, 12.882890375689746644305576446023, 13.16216201361676537487607385719, 13.52129258384090694573451111901, 14.79320736072738101061715713254, 15.13694173825785080780781346051, 16.049317858846983430003524325481, 16.89685864450216884094037824438, 17.702636259455547817940110855806, 18.077890162532360516007171063509

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