↑

Properties

Label 1-4235-4235.1123-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

Related objects

Downloads

Learn more

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: $\chi_{4235} (1123, \cdot )$
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 \( 1 + (-0.755 - 0.654i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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