Properties

Label 1-4235-4235.1167-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.478 + 0.877i$
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  + (−0.618 − 0.786i)2-s + (−0.866 − 0.5i)3-s + (−0.235 + 0.971i)4-s + (0.142 + 0.989i)6-s + (0.909 − 0.415i)8-s + (0.5 + 0.866i)9-s + (0.690 − 0.723i)12-s + (−0.281 − 0.959i)13-s + (−0.888 − 0.458i)16-s + (−0.189 − 0.981i)17-s + (0.371 − 0.928i)18-s + (0.981 + 0.189i)19-s + (−0.458 + 0.888i)23-s + (−0.995 − 0.0950i)24-s + (−0.580 + 0.814i)26-s i·27-s + ⋯
L(s)  = 1  + (−0.618 − 0.786i)2-s + (−0.866 − 0.5i)3-s + (−0.235 + 0.971i)4-s + (0.142 + 0.989i)6-s + (0.909 − 0.415i)8-s + (0.5 + 0.866i)9-s + (0.690 − 0.723i)12-s + (−0.281 − 0.959i)13-s + (−0.888 − 0.458i)16-s + (−0.189 − 0.981i)17-s + (0.371 − 0.928i)18-s + (0.981 + 0.189i)19-s + (−0.458 + 0.888i)23-s + (−0.995 − 0.0950i)24-s + (−0.580 + 0.814i)26-s i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.478 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.478 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.478 + 0.877i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.478 + 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1971192627 + 0.1170347112i\)
\(L(\frac12)\) \(\approx\) \(0.1971192627 + 0.1170347112i\)
\(L(1)\) \(\approx\) \(0.4698192115 - 0.2244353937i\)
\(L(1)\) \(\approx\) \(0.4698192115 - 0.2244353937i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.618 + 0.786i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.281 + 0.959i)T \)
17 \( 1 + (0.189 + 0.981i)T \)
19 \( 1 + (-0.981 - 0.189i)T \)
23 \( 1 + (0.458 - 0.888i)T \)
29 \( 1 + (-0.654 + 0.755i)T \)
31 \( 1 + (0.723 - 0.690i)T \)
37 \( 1 + (0.971 - 0.235i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 + (0.371 + 0.928i)T \)
53 \( 1 + (-0.458 - 0.888i)T \)
59 \( 1 + (0.786 + 0.618i)T \)
61 \( 1 + (0.928 - 0.371i)T \)
67 \( 1 + (0.371 - 0.928i)T \)
71 \( 1 + (0.654 - 0.755i)T \)
73 \( 1 + (0.998 - 0.0475i)T \)
79 \( 1 + (-0.995 + 0.0950i)T \)
83 \( 1 + (0.540 - 0.841i)T \)
89 \( 1 + (0.327 - 0.945i)T \)
97 \( 1 + (-0.909 + 0.415i)T \)
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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

−18.04614105691284371266240667286, −17.50401997961479809789796310534, −16.85947757326784196846678178733, −16.24783557568932464848889048839, −15.8550223530475539212622144609, −15.01846218680245402222845431298, −14.42083072756678737151720068831, −13.7601816272569651825118313831, −12.69017917562346712783643427348, −12.03250616307545534724592929453, −11.18132576412675777360118630475, −10.594592650411217483247352689119, −10.0154597693151158898670057654, −9.16387770898179794947082248683, −8.80711863336987192575728978821, −7.6598882270678791374126129706, −7.0757316896528346474459651959, −6.26128013154201842660792208212, −5.84517588224956357230739353317, −4.87210287273140062692318671259, −4.42634434468526925056568688884, −3.48272219847637231716292312083, −2.06695966772830639415898020424, −1.25274102933245077805958552442, −0.11566027837987634978130099423, 0.89529551977598878254385730072, 1.59082437200078149333137056757, 2.588849351121954496555872052266, 3.25989517820971641182460818461, 4.307335288546759061296876965984, 5.13637596068989375094962844679, 5.74180284564077951468542451183, 6.86023129552470884378911869561, 7.47168543341719026309922058924, 7.958157543069201877484337094129, 8.92315658748478162561600594570, 9.77951200771775238914528041826, 10.27949174160848527461427651227, 10.99688890377752164200366762308, 11.76100220693684989448969973342, 12.07484139848253974326633782071, 12.839190349082175005132827977466, 13.56458648985606505339306431776, 14.034357107647645085678324585947, 15.43309315353336598528032209665, 16.00183238711423302683951440493, 16.63836878373832332700601153529, 17.44625649395640993702225046115, 17.8956667557668109965679287618, 18.31276625170490745053922768480

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