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Properties

Label 1-4235-4235.1052-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.999 - 0.0379i$
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  + (0.299 − 0.953i)2-s + (−0.406 − 0.913i)3-s + (−0.820 − 0.572i)4-s + (−0.993 + 0.113i)6-s + (−0.791 + 0.610i)8-s + (−0.669 + 0.743i)9-s + (−0.189 + 0.981i)12-s + (0.389 − 0.921i)13-s + (0.345 + 0.938i)16-s + (0.263 − 0.964i)17-s + (0.508 + 0.861i)18-s + (−0.625 + 0.780i)19-s + (0.618 + 0.786i)23-s + (0.879 + 0.475i)24-s + (−0.761 − 0.647i)26-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (0.299 − 0.953i)2-s + (−0.406 − 0.913i)3-s + (−0.820 − 0.572i)4-s + (−0.993 + 0.113i)6-s + (−0.791 + 0.610i)8-s + (−0.669 + 0.743i)9-s + (−0.189 + 0.981i)12-s + (0.389 − 0.921i)13-s + (0.345 + 0.938i)16-s + (0.263 − 0.964i)17-s + (0.508 + 0.861i)18-s + (−0.625 + 0.780i)19-s + (0.618 + 0.786i)23-s + (0.879 + 0.475i)24-s + (−0.761 − 0.647i)26-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.999 - 0.0379i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1052, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.999 - 0.0379i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02415125886 - 1.272244402i\)
\(L(\frac12)\) \(\approx\) \(0.02415125886 - 1.272244402i\)
\(L(1)\) \(\approx\) \(0.6167476487 - 0.7154217711i\)
\(L(1)\) \(\approx\) \(0.6167476487 - 0.7154217711i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.299 - 0.953i)T \)
3 \( 1 + (-0.406 - 0.913i)T \)
13 \( 1 + (0.389 - 0.921i)T \)
17 \( 1 + (0.263 - 0.964i)T \)
19 \( 1 + (-0.625 + 0.780i)T \)
23 \( 1 + (0.618 + 0.786i)T \)
29 \( 1 + (-0.998 + 0.0570i)T \)
31 \( 1 + (0.483 + 0.875i)T \)
37 \( 1 + (0.956 - 0.290i)T \)
41 \( 1 + (-0.198 + 0.980i)T \)
43 \( 1 + (0.281 - 0.959i)T \)
47 \( 1 + (-0.508 + 0.861i)T \)
53 \( 1 + (-0.938 - 0.345i)T \)
59 \( 1 + (0.948 + 0.318i)T \)
61 \( 1 + (0.217 - 0.976i)T \)
67 \( 1 + (0.0950 - 0.995i)T \)
71 \( 1 + (0.774 - 0.633i)T \)
73 \( 1 + (-0.244 - 0.969i)T \)
79 \( 1 + (-0.432 + 0.901i)T \)
83 \( 1 + (-0.441 + 0.897i)T \)
89 \( 1 + (0.888 + 0.458i)T \)
97 \( 1 + (0.999 - 0.0285i)T \)
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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

−18.64888291286653881821872986356, −17.67042428313551215114590732836, −17.075077087079960735846074960339, −16.70750766996422324451314204337, −15.98368763836323603024992106058, −15.38041987118851883987813351292, −14.66224072566008931712852518417, −14.39767054636420840388817246176, −13.19295748194897033594706022190, −12.88597576290872748549080704782, −11.76597312956220284439282014172, −11.286832280909989819671846993462, −10.37100442086471571312400771716, −9.65115458504892133480975871608, −8.88291056649393296616660631853, −8.50103467433186001946338324124, −7.487905526460022838071019901615, −6.56259745362597121267608314556, −6.15271962199158236894810259125, −5.373151127799586278834342948932, −4.514104171829699253292824201437, −4.127092070359340354705927963804, −3.34934592687849142674278560727, −2.30959264334637941523439884161, −0.80771859177734295717121980424, 0.472256584181712127185155284391, 1.2919279981666972516298402419, 2.02393922885127942725458150535, 2.95574152627659288698836703912, 3.50046655210012425193302610069, 4.66777033907216904623653648348, 5.34594835919144062692453204810, 5.928205534017678504572664722341, 6.732805233931518764616051877712, 7.74669978770864275495618369140, 8.24651095864361761583485343123, 9.19991576966948409946125262749, 9.906761853173905280957212221004, 10.81701174612986589679920095590, 11.20132541504258974256412685670, 11.94900386303197076998211074707, 12.64237038776506846249560001675, 13.05874599923031653419245530924, 13.75027073555301146418309717508, 14.39102696932052671326745595315, 15.12807364601290638894423241404, 16.06308533108743007680566705899, 16.95408976257968022810078017884, 17.58756188451247682689192132564, 18.258167481356593058386556052467

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