Properties

Label 1-4235-4235.3477-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.545 - 0.838i$
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  + (−0.971 + 0.235i)2-s + (−0.866 − 0.5i)3-s + (0.888 − 0.458i)4-s + (0.959 + 0.281i)6-s + (−0.755 + 0.654i)8-s + (0.5 + 0.866i)9-s + (−0.998 − 0.0475i)12-s + (−0.540 + 0.841i)13-s + (0.580 − 0.814i)16-s + (0.371 − 0.928i)17-s + (−0.690 − 0.723i)18-s + (0.928 − 0.371i)19-s + (−0.814 − 0.580i)23-s + (0.981 − 0.189i)24-s + (0.327 − 0.945i)26-s i·27-s + ⋯
L(s)  = 1  + (−0.971 + 0.235i)2-s + (−0.866 − 0.5i)3-s + (0.888 − 0.458i)4-s + (0.959 + 0.281i)6-s + (−0.755 + 0.654i)8-s + (0.5 + 0.866i)9-s + (−0.998 − 0.0475i)12-s + (−0.540 + 0.841i)13-s + (0.580 − 0.814i)16-s + (0.371 − 0.928i)17-s + (−0.690 − 0.723i)18-s + (0.928 − 0.371i)19-s + (−0.814 − 0.580i)23-s + (0.981 − 0.189i)24-s + (0.327 − 0.945i)26-s i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.545 - 0.838i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (3477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.545 - 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2503821509 - 0.4617753243i\)
\(L(\frac12)\) \(\approx\) \(0.2503821509 - 0.4617753243i\)
\(L(1)\) \(\approx\) \(0.5137445291 - 0.1015160784i\)
\(L(1)\) \(\approx\) \(0.5137445291 - 0.1015160784i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.971 + 0.235i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-0.540 + 0.841i)T \)
17 \( 1 + (0.371 - 0.928i)T \)
19 \( 1 + (0.928 - 0.371i)T \)
23 \( 1 + (-0.814 - 0.580i)T \)
29 \( 1 + (0.142 - 0.989i)T \)
31 \( 1 + (-0.0475 - 0.998i)T \)
37 \( 1 + (0.458 - 0.888i)T \)
41 \( 1 + (0.959 + 0.281i)T \)
43 \( 1 + (-0.755 + 0.654i)T \)
47 \( 1 + (0.690 - 0.723i)T \)
53 \( 1 + (0.814 - 0.580i)T \)
59 \( 1 + (0.235 - 0.971i)T \)
61 \( 1 + (-0.723 - 0.690i)T \)
67 \( 1 + (0.690 + 0.723i)T \)
71 \( 1 + (-0.142 + 0.989i)T \)
73 \( 1 + (0.0950 - 0.995i)T \)
79 \( 1 + (-0.981 - 0.189i)T \)
83 \( 1 + (0.909 + 0.415i)T \)
89 \( 1 + (-0.786 + 0.618i)T \)
97 \( 1 + (-0.755 + 0.654i)T \)
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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

−18.29263809566938544800406688419, −17.95762560941317225552564019990, −17.2963612295449541565016359088, −16.6747433195289957531898863268, −16.13016228829239323056986243403, −15.41258441976513985249934631956, −14.91469993766485897979471331789, −13.89327931854217648139419530917, −12.6938849963451077324647288374, −12.30121496581543170424474074010, −11.70652408018553195868454891347, −10.856160721225070022830961983394, −10.31218535676854525175420883617, −9.87913331149143378909906117104, −9.09595813713314658511468024985, −8.27277063889547463724396006539, −7.4993044663331757646388030866, −6.892533371655138655597588652688, −5.85268436186658184209536950033, −5.53930112641524149079708681138, −4.397931644957525802465590572211, −3.504265220739907310351657113603, −2.889697296299244223448903634949, −1.60648579884306972787186443992, −0.97012703700309564337738728206, 0.303480330449652919680356106691, 1.05138434496741071671402884153, 2.131901559702156752067960612874, 2.59607243758476706161504095718, 4.041508110939655055957823182420, 4.9878187920978944713003997462, 5.6661084809315658601093140798, 6.40158748768399163580779645285, 7.04941743059810430353017791439, 7.644556986295118409161336860353, 8.23807095223101189851444928943, 9.47948345296237732976365887258, 9.660046139565090223274319544682, 10.57947692607700191151118912423, 11.442465731566277247859857466523, 11.71769823479332844795442898607, 12.386469025231763535091481626244, 13.40464021783819715688582245174, 14.12226875925312142048763637808, 14.84682143721682237702421697510, 15.90794784371492590959819809841, 16.21171252993374428100885040139, 16.878134437036928660061754200084, 17.48457422011457013570298238404, 18.19853164461678452207577559502

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