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Properties

Label 1-4235-4235.4078-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: $\chi_{4235} (4078, \cdot )$
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

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

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