↑

Properties

Label 1-4235-4235.2817-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.993 + 0.113i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.993 + 0.113i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2817, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.993 + 0.113i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.540490741 + 0.3146312018i\)
\(L(\frac12)\) \(\approx\) \(5.540490741 + 0.3146312018i\)
\(L(1)\) \(\approx\) \(2.789629072 + 0.09082183523i\)
\(L(1)\) \(\approx\) \(2.789629072 + 0.09082183523i\)

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

−18.73014450409342063651561905222, −17.568027771774462407965903214563, −16.88067548375425110523123636685, −15.817266497387614793197273703920, −15.56094368980059491458321937106, −15.02110131107040111708137398181, −14.212337144946426031618880725158, −13.634807172126396152487806492016, −13.04223657793355889141164522347, −12.53810361086848753832001591188, −11.296622769713140263891686248654, −11.064040244008532174137177551572, −10.088765683650118021644700824400, −9.61697563916429752248601296017, −8.66496745188316309692470868107, −7.84376233775121369475598436838, −7.30801874603571425497909421642, −6.22903035673202668615360498113, −5.60498898773316673239595503805, −4.73403062076945727990026107328, −4.13007265336236610357702619208, −3.29771480687190910543022502828, −2.865727251282712082288183369310, −1.92471966154870286007059515296, −1.075560455527838293771997538496, 1.14949430659810092579657787685, 1.86682172441105452969330396038, 2.833206565051499200393520467618, 3.27587013318162724344670421321, 4.1759587855365008351903605275, 4.854256121002509286345412716890, 5.77481402601133241861708774373, 6.63001722061470179852221473887, 7.15696701659994983456513953659, 7.68591965814443557567996182818, 8.78748365382229999316826491569, 9.05780921547341025854385396177, 10.21097840686198483549414373793, 11.08929722787852809692550203959, 11.77593910848203636683452860242, 12.449624095472176574985698940954, 13.066844649715646005862319450524, 13.74685652436070159327937623204, 14.32017388400357589626307348946, 14.62924920164395479838465460226, 15.73615023741536916558594790854, 16.04875376710565656904711332423, 16.776753674681738739786097660308, 17.973846802838967492265183923540, 18.22652232693376575008097795317

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