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Properties

Label 1-4235-4235.1783-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

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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

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

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: $\chi_{4235} (1783, \cdot )$
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 \( 1 + (0.755 - 0.654i)T \)
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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

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

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