Properties

Label 1-4235-4235.1229-r1-0-0
Degree $1$
Conductor $4235$
Sign $0.744 + 0.667i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (−0.851 − 0.524i)2-s + (−0.913 − 0.406i)3-s + (0.449 + 0.893i)4-s + (0.564 + 0.825i)6-s + (0.0855 − 0.996i)8-s + (0.669 + 0.743i)9-s + (−0.0475 − 0.998i)12-s + (−0.254 − 0.967i)13-s + (−0.595 + 0.803i)16-s + (0.640 + 0.768i)17-s + (−0.179 − 0.983i)18-s + (0.969 − 0.244i)19-s + (−0.580 − 0.814i)23-s + (−0.483 + 0.875i)24-s + (−0.290 + 0.956i)26-s + (−0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (−0.851 − 0.524i)2-s + (−0.913 − 0.406i)3-s + (0.449 + 0.893i)4-s + (0.564 + 0.825i)6-s + (0.0855 − 0.996i)8-s + (0.669 + 0.743i)9-s + (−0.0475 − 0.998i)12-s + (−0.254 − 0.967i)13-s + (−0.595 + 0.803i)16-s + (0.640 + 0.768i)17-s + (−0.179 − 0.983i)18-s + (0.969 − 0.244i)19-s + (−0.580 − 0.814i)23-s + (−0.483 + 0.875i)24-s + (−0.290 + 0.956i)26-s + (−0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.744 + 0.667i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ 0.744 + 0.667i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3083833420 + 0.1180363008i\)
\(L(\frac12)\) \(\approx\) \(0.3083833420 + 0.1180363008i\)
\(L(1)\) \(\approx\) \(0.4723393205 - 0.1840491709i\)
\(L(1)\) \(\approx\) \(0.4723393205 - 0.1840491709i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.851 - 0.524i)T \)
3 \( 1 + (-0.913 - 0.406i)T \)
13 \( 1 + (-0.254 - 0.967i)T \)
17 \( 1 + (0.640 + 0.768i)T \)
19 \( 1 + (0.969 - 0.244i)T \)
23 \( 1 + (-0.580 - 0.814i)T \)
29 \( 1 + (0.466 - 0.884i)T \)
31 \( 1 + (-0.935 - 0.353i)T \)
37 \( 1 + (-0.988 - 0.151i)T \)
41 \( 1 + (-0.610 + 0.791i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (0.179 - 0.983i)T \)
53 \( 1 + (0.595 + 0.803i)T \)
59 \( 1 + (0.380 - 0.924i)T \)
61 \( 1 + (-0.879 - 0.475i)T \)
67 \( 1 + (-0.723 - 0.690i)T \)
71 \( 1 + (0.897 - 0.441i)T \)
73 \( 1 + (0.861 + 0.508i)T \)
79 \( 1 + (-0.123 + 0.992i)T \)
83 \( 1 + (-0.736 + 0.676i)T \)
89 \( 1 + (-0.786 - 0.618i)T \)
97 \( 1 + (-0.516 - 0.856i)T \)
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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

−18.0110052755973772385901611311, −17.47386086218509402628599110755, −16.610494384639510561063038050078, −16.3071994393856461758837975746, −15.74030621475835572558107811337, −14.96374618661288455459655544716, −14.177153303205590055116088356626, −13.6385557354165525612785774504, −12.205699916990220608592336472, −11.91717799591663165805572822284, −11.18972851431928628690202641107, −10.41483843816252012736962528811, −9.84025881633185645532715891023, −9.25611729765325824742337259376, −8.579740720396553076336046737422, −7.32357693898576981188352811182, −7.17929776380688254071218224878, −6.28185955783950573778138474568, −5.32550771210174078457419703646, −5.19199404008907462409433645684, −4.02482886749158823151462415226, −3.103240760096097811003153105194, −1.789449411229792246455393428449, −1.18376081766081898793695263604, −0.12428322239775070080489876900, 0.581741127593081291833539125828, 1.37766326349483215345607740382, 2.17893180976093032733684767483, 3.10573157216754629263079432342, 3.940020774821915205905855026165, 4.94932515955477341963344381138, 5.73746965130915421752422426044, 6.49455967058814280557800905084, 7.25096705259820960991386552147, 7.954780410285013514690412914, 8.40420290887452194052513544692, 9.63305383093031740450611481411, 10.0881544451337537469913498592, 10.71866938446629649890435462816, 11.386834408918841376115618610716, 12.12615376690379340208107795259, 12.5327344982288147153709169621, 13.22383064607145280209808673876, 14.02953335775215040487259278716, 15.233429222975307474601455547492, 15.727674901717452239958574619, 16.75905537605965959802332895866, 16.8518376855997446367337520947, 17.753817683609229271030662342500, 18.31124418191184690245654957461

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