↑

Properties

Label 1-4235-4235.1264-r1-0-0
Degree $1$
Conductor $4235$
Sign $-0.947 - 0.320i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0475 + 0.998i)2-s + (0.5 + 0.866i)3-s + (−0.995 + 0.0950i)4-s + (−0.841 + 0.540i)6-s + (−0.142 − 0.989i)8-s + (−0.5 + 0.866i)9-s + (−0.580 − 0.814i)12-s + (0.415 − 0.909i)13-s + (0.981 − 0.189i)16-s + (0.235 + 0.971i)17-s + (−0.888 − 0.458i)18-s + (−0.235 + 0.971i)19-s + (−0.981 + 0.189i)23-s + (0.786 − 0.618i)24-s + (0.928 + 0.371i)26-s − 27-s + ⋯
L(s)  = 1  + (0.0475 + 0.998i)2-s + (0.5 + 0.866i)3-s + (−0.995 + 0.0950i)4-s + (−0.841 + 0.540i)6-s + (−0.142 − 0.989i)8-s + (−0.5 + 0.866i)9-s + (−0.580 − 0.814i)12-s + (0.415 − 0.909i)13-s + (0.981 − 0.189i)16-s + (0.235 + 0.971i)17-s + (−0.888 − 0.458i)18-s + (−0.235 + 0.971i)19-s + (−0.981 + 0.189i)23-s + (0.786 − 0.618i)24-s + (0.928 + 0.371i)26-s − 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.947 - 0.320i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1264, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ -0.947 - 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3368147659 + 2.047514033i\)
\(L(\frac12)\) \(\approx\) \(-0.3368147659 + 2.047514033i\)
\(L(1)\) \(\approx\) \(0.7452888369 + 0.8540634712i\)
\(L(1)\) \(\approx\) \(0.7452888369 + 0.8540634712i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.0475 + 0.998i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.415 - 0.909i)T \)
17 \( 1 + (0.235 + 0.971i)T \)
19 \( 1 + (-0.235 + 0.971i)T \)
23 \( 1 + (-0.981 + 0.189i)T \)
29 \( 1 + (0.959 - 0.281i)T \)
31 \( 1 + (0.580 - 0.814i)T \)
37 \( 1 + (0.995 + 0.0950i)T \)
41 \( 1 + (-0.841 + 0.540i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (0.888 - 0.458i)T \)
53 \( 1 + (-0.981 - 0.189i)T \)
59 \( 1 + (0.0475 - 0.998i)T \)
61 \( 1 + (0.888 - 0.458i)T \)
67 \( 1 + (0.888 + 0.458i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (-0.327 + 0.945i)T \)
79 \( 1 + (0.786 + 0.618i)T \)
83 \( 1 + (-0.654 + 0.755i)T \)
89 \( 1 + (0.723 - 0.690i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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

−17.91286638832942671183552423194, −17.654087017936984111946033159471, −16.585626919378620072388821387152, −15.78435789494051517801503111548, −14.78695666711483506675301586947, −14.07441917449842336761563226681, −13.74377729754957594158834417341, −13.080971451958955383189517914590, −12.25020251124476609980907315599, −11.8152732638402248099715089250, −11.18884304640641346236663040723, −10.29228616660832676118030775003, −9.46228012943800373162696616707, −8.91548364470467591641821547258, −8.30725331238893329518365425452, −7.48202031096595157920753416182, −6.64231149075933494843683209450, −5.95113575478607590826664642197, −4.83961529398239568457681884407, −4.235890709653981662091068764577, −3.21621997769085124468231680390, −2.67811427544344521214055280036, −1.90320265966690984590484157173, −1.107074848741165952264476493869, −0.36490158654789286766395460851, 0.78403460900041651724339308097, 2.03316557159731041980894579029, 3.15295721846437021665431210312, 3.84774894476312370856247653759, 4.3462327305867276773636019507, 5.34506992146951379704686734019, 5.861555311975044364443950336985, 6.564391541559914559482154040797, 7.78596161423630260928783622561, 8.19047850383288184262485155444, 8.58395322760813592475197176019, 9.770170734195243785290699152143, 10.00297984751480752887388125407, 10.74384305107782041698775008449, 11.81782050086914456343322985447, 12.7097823753232568166631574589, 13.36177222626469116356119005686, 14.08716353443439112273312695598, 14.61959984781034403221922325160, 15.32898389818808471001212321704, 15.7236538754603026872763911690, 16.42664863290309196890041720913, 17.12483867783658834416521590346, 17.59194098901538435845999554669, 18.61713437782619988818051940207

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