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Properties

Label 1-4235-4235.1132-r1-0-0
Degree $1$
Conductor $4235$
Sign $0.877 - 0.478i$
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  + (0.618 + 0.786i)2-s + (−0.866 − 0.5i)3-s + (−0.235 + 0.971i)4-s + (−0.142 − 0.989i)6-s + (−0.909 + 0.415i)8-s + (0.5 + 0.866i)9-s + (0.690 − 0.723i)12-s + (0.281 + 0.959i)13-s + (−0.888 − 0.458i)16-s + (0.189 + 0.981i)17-s + (−0.371 + 0.928i)18-s + (−0.981 − 0.189i)19-s + (−0.458 + 0.888i)23-s + (0.995 + 0.0950i)24-s + (−0.580 + 0.814i)26-s − i·27-s + ⋯
L(s)  = 1  + (0.618 + 0.786i)2-s + (−0.866 − 0.5i)3-s + (−0.235 + 0.971i)4-s + (−0.142 − 0.989i)6-s + (−0.909 + 0.415i)8-s + (0.5 + 0.866i)9-s + (0.690 − 0.723i)12-s + (0.281 + 0.959i)13-s + (−0.888 − 0.458i)16-s + (0.189 + 0.981i)17-s + (−0.371 + 0.928i)18-s + (−0.981 − 0.189i)19-s + (−0.458 + 0.888i)23-s + (0.995 + 0.0950i)24-s + (−0.580 + 0.814i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.877 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.877 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.877 - 0.478i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ 0.877 - 0.478i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06625661488 + 0.01689022494i\)
\(L(\frac12)\) \(\approx\) \(-0.06625661488 + 0.01689022494i\)
\(L(1)\) \(\approx\) \(0.6846837815 + 0.4717378928i\)
\(L(1)\) \(\approx\) \(0.6846837815 + 0.4717378928i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.618 + 0.786i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (0.281 + 0.959i)T \)
17 \( 1 + (0.189 + 0.981i)T \)
19 \( 1 + (-0.981 - 0.189i)T \)
23 \( 1 + (-0.458 + 0.888i)T \)
29 \( 1 + (-0.654 + 0.755i)T \)
31 \( 1 + (-0.723 + 0.690i)T \)
37 \( 1 + (-0.971 + 0.235i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 + (-0.371 - 0.928i)T \)
53 \( 1 + (0.458 + 0.888i)T \)
59 \( 1 + (-0.786 - 0.618i)T \)
61 \( 1 + (0.928 - 0.371i)T \)
67 \( 1 + (-0.371 + 0.928i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (0.998 - 0.0475i)T \)
79 \( 1 + (-0.995 + 0.0950i)T \)
83 \( 1 + (0.540 - 0.841i)T \)
89 \( 1 + (-0.327 + 0.945i)T \)
97 \( 1 + (0.909 - 0.415i)T \)
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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

−17.74948126125721537962607120557, −16.83457987475753982509996418336, −16.2167394994254084854040490971, −15.37620811776539930267846873957, −14.95249617201790238288820239164, −14.188238041560586652572314029200, −13.21231734513043118036947224198, −12.79346415597696979018891913630, −11.99998843479851912869624048918, −11.47733373965169124379975486910, −10.736801259119022417504505817774, −10.2676120768800629262968950807, −9.60443312072989552733440101212, −8.851739510780248120151589281824, −7.838362389660574963508454864516, −6.72757816999430403304512691040, −6.10784910504669921898387435593, −5.43631320139404418056310519349, −4.81162819910892940740463225494, −4.05446935644344600033293278813, −3.40609192188748654218149044126, −2.49581976375470245357554623228, −1.54204717153939504026923265750, −0.48598590201934353871058023035, −0.01559147304010988753763102726, 1.5094452769083506455511637852, 2.09501042182256530685081899547, 3.466838508597955143851580049832, 4.055297230179243357449680562784, 4.92391233831397895699037678765, 5.57534519368278435958152839051, 6.220543151635563660823534512542, 6.908954833552136857192224977725, 7.349919239725334748329899756551, 8.370981618787433712774094384460, 8.8326993740790647004419600567, 9.956974995462353422243432879525, 10.84872891154369675948550134668, 11.46687460215862573412815541902, 12.2113396958320284085227053817, 12.7195613140816728660195257401, 13.399597560384494971743672598432, 14.01089130988032412314572408191, 14.783734224561666609349692602414, 15.50746929616475544333962850015, 16.22854024536442225773274018577, 16.78264309461732840159067467967, 17.30575635856458665261953271696, 17.92262350841279084571847376780, 18.6938555655702193765448679643

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