Properties

Label 1-4235-4235.758-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.564 + 0.825i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.564 + 0.825i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (758, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.564 + 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2021237602 + 0.3833050338i\)
\(L(\frac12)\) \(\approx\) \(0.2021237602 + 0.3833050338i\)
\(L(1)\) \(\approx\) \(0.5255155067 + 0.04503527352i\)
\(L(1)\) \(\approx\) \(0.5255155067 + 0.04503527352i\)

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

−17.971534105685453932701053927988, −17.61640875199141170729803597566, −16.62217783929981784647809593592, −16.44331959573369786447405904311, −15.6709716513306455985242841007, −15.13519093889971262915341028947, −14.10218105586747196879508728260, −13.24352207020679577654928915009, −12.28778041695994201516000163649, −11.81328412019874727032568068284, −11.102790112026667914041123664797, −10.76149055653673623279775001575, −9.698249650825783584219585453961, −9.40478557304810025366213525281, −8.6970276099105256246556807928, −7.636579133655879296119686022702, −7.04159167877775804095819679692, −6.28868765177706960190870732696, −5.65528875476178679141167243611, −4.63124205244253950715800138207, −3.89596471771649765301575897888, −3.033984315623140275332344962992, −2.0522964113797037580785840103, −1.098318035417889385842099106191, −0.22679856924922315531085377342, 1.194916353914523088579923689409, 1.373691858820427257265526607075, 2.65453946751115772781739046425, 3.44117726210425126470106850640, 4.83736583916046785921977932696, 5.447504507203992041602085817939, 6.14627355508009859319530766205, 6.8251116667982076912388104947, 7.45288309814028513942424933991, 8.15100617505556475219235752988, 8.85689297241774163645877285378, 9.68654539185974099306194607691, 10.634832699873047087696540344739, 10.80718260602539722444898212799, 11.6369778647121470419281828467, 12.329581891336905107482374175443, 13.05077052877559654926090360648, 13.76941361095267747903297184106, 14.79012473185022545564854876605, 15.53394542457513696185312929525, 16.003792095815272324030321082983, 16.76725836609270444970034502458, 17.40442275051195535873150057885, 17.848680000941790710615925100623, 18.44797970936284810079385405437

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