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Properties

Label 1-4235-4235.3244-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.710 - 0.703i$
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  + (0.981 − 0.189i)2-s + (−0.5 − 0.866i)3-s + (0.928 − 0.371i)4-s + (−0.654 − 0.755i)6-s + (0.841 − 0.540i)8-s + (−0.5 + 0.866i)9-s + (−0.786 − 0.618i)12-s + (0.142 − 0.989i)13-s + (0.723 − 0.690i)16-s + (−0.580 + 0.814i)17-s + (−0.327 + 0.945i)18-s + (0.580 + 0.814i)19-s + (−0.723 + 0.690i)23-s + (−0.888 − 0.458i)24-s + (−0.0475 − 0.998i)26-s + 27-s + ⋯
L(s)  = 1  + (0.981 − 0.189i)2-s + (−0.5 − 0.866i)3-s + (0.928 − 0.371i)4-s + (−0.654 − 0.755i)6-s + (0.841 − 0.540i)8-s + (−0.5 + 0.866i)9-s + (−0.786 − 0.618i)12-s + (0.142 − 0.989i)13-s + (0.723 − 0.690i)16-s + (−0.580 + 0.814i)17-s + (−0.327 + 0.945i)18-s + (0.580 + 0.814i)19-s + (−0.723 + 0.690i)23-s + (−0.888 − 0.458i)24-s + (−0.0475 − 0.998i)26-s + 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.710 - 0.703i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (3244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.710 - 0.703i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9007692806 - 2.190839528i\)
\(L(\frac12)\) \(\approx\) \(0.9007692806 - 2.190839528i\)
\(L(1)\) \(\approx\) \(1.372293171 - 0.7661919508i\)
\(L(1)\) \(\approx\) \(1.372293171 - 0.7661919508i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.981 - 0.189i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.142 - 0.989i)T \)
17 \( 1 + (-0.580 + 0.814i)T \)
19 \( 1 + (0.580 + 0.814i)T \)
23 \( 1 + (-0.723 + 0.690i)T \)
29 \( 1 + (-0.415 + 0.909i)T \)
31 \( 1 + (0.786 - 0.618i)T \)
37 \( 1 + (-0.928 - 0.371i)T \)
41 \( 1 + (-0.654 - 0.755i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (-0.327 - 0.945i)T \)
53 \( 1 + (-0.723 - 0.690i)T \)
59 \( 1 + (-0.981 - 0.189i)T \)
61 \( 1 + (-0.327 - 0.945i)T \)
67 \( 1 + (0.327 - 0.945i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (-0.235 - 0.971i)T \)
79 \( 1 + (0.888 - 0.458i)T \)
83 \( 1 + (0.959 - 0.281i)T \)
89 \( 1 + (0.995 + 0.0950i)T \)
97 \( 1 + (0.841 - 0.540i)T \)
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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

−18.57735961295680034320459081949, −17.63628005472758863642609204966, −17.14396241946692797996501597051, −16.27396859250248814892752718818, −15.90537178003706102000928241668, −15.38575813101213330703270031959, −14.51805357445041883925678035097, −13.95473980903092468640090163834, −13.376246690923461053058277380487, −12.34609474503748094305150628722, −11.75988203968084889537991950494, −11.303674855594376279267546108829, −10.61345090697264795742541316565, −9.74980000328666454162182895265, −9.076227663807777532193737618488, −8.201191052265019238894108979063, −7.190749823703001414084578398337, −6.504500525661468542828107481993, −5.99685549515498650434619788401, −4.9762046595753899584639518219, −4.601861759012823987677783109725, −3.950040915640686553082907389929, −3.02251555425445457379197438750, −2.373799734803471614382058583870, −1.1314543405704732492594300723, 0.48158012314699792986336981203, 1.689188431400495108174693083593, 2.01305956098996766262852223324, 3.24010519889133140639399151942, 3.72317481291223240071305872404, 4.92564253417340198758042980179, 5.43496118003894805522427084068, 6.1496731193751714861938027931, 6.65742860266956200145227101730, 7.720562603463750934230651850871, 7.92421537692230877585833886371, 9.14719144452846268348037807018, 10.40327055999866212705899791593, 10.590713333430341659251737363405, 11.5249460857694226674247590842, 12.16070512474317485898394708161, 12.61246500228528790536885325323, 13.35441192083426610853030850781, 13.830228213287133627998709338836, 14.55340203987767526480678048239, 15.429328356310290924615142295032, 15.90923361162400416554399381884, 16.815303828759066995045074214715, 17.40069639851057793268481697681, 18.15602529370032247472064302123

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