Properties

Label 1-4235-4235.479-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.868 - 0.495i$
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  + (0.999 + 0.0380i)2-s + (0.669 − 0.743i)3-s + (0.997 + 0.0760i)4-s + (0.696 − 0.717i)6-s + (0.993 + 0.113i)8-s + (−0.104 − 0.994i)9-s + (0.723 − 0.690i)12-s + (−0.941 + 0.336i)13-s + (0.988 + 0.151i)16-s + (0.683 + 0.730i)17-s + (−0.0665 − 0.997i)18-s + (0.483 + 0.875i)19-s + (0.888 − 0.458i)23-s + (0.749 − 0.662i)24-s + (−0.953 + 0.299i)26-s + (−0.809 − 0.587i)27-s + ⋯
L(s)  = 1  + (0.999 + 0.0380i)2-s + (0.669 − 0.743i)3-s + (0.997 + 0.0760i)4-s + (0.696 − 0.717i)6-s + (0.993 + 0.113i)8-s + (−0.104 − 0.994i)9-s + (0.723 − 0.690i)12-s + (−0.941 + 0.336i)13-s + (0.988 + 0.151i)16-s + (0.683 + 0.730i)17-s + (−0.0665 − 0.997i)18-s + (0.483 + 0.875i)19-s + (0.888 − 0.458i)23-s + (0.749 − 0.662i)24-s + (−0.953 + 0.299i)26-s + (−0.809 − 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.868 - 0.495i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.868 - 0.495i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.877650220 - 1.292369196i\)
\(L(\frac12)\) \(\approx\) \(4.877650220 - 1.292369196i\)
\(L(1)\) \(\approx\) \(2.550990706 - 0.4781475833i\)
\(L(1)\) \(\approx\) \(2.550990706 - 0.4781475833i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.999 + 0.0380i)T \)
3 \( 1 + (0.669 - 0.743i)T \)
13 \( 1 + (-0.941 + 0.336i)T \)
17 \( 1 + (0.683 + 0.730i)T \)
19 \( 1 + (0.483 + 0.875i)T \)
23 \( 1 + (0.888 - 0.458i)T \)
29 \( 1 + (0.921 - 0.389i)T \)
31 \( 1 + (0.991 + 0.132i)T \)
37 \( 1 + (0.851 + 0.524i)T \)
41 \( 1 + (-0.985 + 0.170i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (-0.0665 + 0.997i)T \)
53 \( 1 + (-0.988 + 0.151i)T \)
59 \( 1 + (-0.345 - 0.938i)T \)
61 \( 1 + (-0.532 - 0.846i)T \)
67 \( 1 + (-0.928 + 0.371i)T \)
71 \( 1 + (0.0855 + 0.996i)T \)
73 \( 1 + (0.935 + 0.353i)T \)
79 \( 1 + (-0.861 + 0.508i)T \)
83 \( 1 + (0.998 + 0.0570i)T \)
89 \( 1 + (0.327 + 0.945i)T \)
97 \( 1 + (0.198 + 0.980i)T \)
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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

−18.62540066495302157996682788452, −17.56172981740256422664493495319, −16.80018075606529369859047431764, −16.18120196470184381109094661524, −15.53823696307898870525935126673, −14.953326532947027611731720954308, −14.456880103517135306449679819811, −13.64909866812470290432740197061, −13.30008196178850638807785786580, −12.289242873431539062862476291942, −11.70894636156630073188923067334, −10.92519958936827376475881918222, −10.21197833435603824552744816776, −9.613422585222594914446341562842, −8.842288764987045407877222738275, −7.73877717646810233711244107852, −7.413571304458079265059977859307, −6.4612449283994251821027025781, −5.42900259078121877901091862488, −4.86533911043723729337610034832, −4.4370045043205021982813001299, −3.16623629280087638117572086396, −3.04123122535071181329315538693, −2.183490317328244457490539904625, −1.0139248734777785288562458841, 1.06068962829939460553021368555, 1.76905340484096745294212042979, 2.7071942308126309309488476649, 3.15487415600848149077403039653, 4.08512374607293418372095021504, 4.82048067432236842916027946437, 5.72502372340518749958309907703, 6.494619424095214580464294377239, 6.978066881524691645828931394410, 7.98108506590629925414887374277, 8.14615325835329692464493003953, 9.437831444745483560275089941870, 10.06004292354799699866330263206, 10.9470664920712115266388827556, 11.97597319557232247030561206932, 12.2245624526764621565918185485, 12.87263006860743083523582397558, 13.67082043144171648240058484252, 14.21349965391867835065587123161, 14.72481282118734657185642954694, 15.324591750571459253470988550272, 16.128690817071471222683716415439, 17.06763736056388221650975315088, 17.38181288335927460256648196102, 18.691076903626207117870967978584

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