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Properties

Label 1-4235-4235.3979-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  + (−0.830 + 0.556i)2-s + (0.913 + 0.406i)3-s + (0.380 − 0.924i)4-s + (−0.985 + 0.170i)6-s + (0.198 + 0.980i)8-s + (0.669 + 0.743i)9-s + (0.723 − 0.690i)12-s + (0.564 − 0.825i)13-s + (−0.710 − 0.703i)16-s + (−0.123 − 0.992i)17-s + (−0.969 − 0.244i)18-s + (−0.683 + 0.730i)19-s + (0.888 − 0.458i)23-s + (−0.217 + 0.976i)24-s + (−0.00951 + 0.999i)26-s + (0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (−0.830 + 0.556i)2-s + (0.913 + 0.406i)3-s + (0.380 − 0.924i)4-s + (−0.985 + 0.170i)6-s + (0.198 + 0.980i)8-s + (0.669 + 0.743i)9-s + (0.723 − 0.690i)12-s + (0.564 − 0.825i)13-s + (−0.710 − 0.703i)16-s + (−0.123 − 0.992i)17-s + (−0.969 − 0.244i)18-s + (−0.683 + 0.730i)19-s + (0.888 − 0.458i)23-s + (−0.217 + 0.976i)24-s + (−0.00951 + 0.999i)26-s + (0.309 + 0.951i)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: $\chi_{4235} (3979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.868 - 0.495i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.393172632 - 0.3691313058i\)
\(L(\frac12)\) \(\approx\) \(1.393172632 - 0.3691313058i\)
\(L(1)\) \(\approx\) \(0.9790040153 + 0.1668498509i\)
\(L(1)\) \(\approx\) \(0.9790040153 + 0.1668498509i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.830 + 0.556i)T \)
3 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (0.564 - 0.825i)T \)
17 \( 1 + (-0.123 - 0.992i)T \)
19 \( 1 + (-0.683 + 0.730i)T \)
23 \( 1 + (0.888 - 0.458i)T \)
29 \( 1 + (-0.0855 - 0.996i)T \)
31 \( 1 + (-0.879 + 0.475i)T \)
37 \( 1 + (0.761 - 0.647i)T \)
41 \( 1 + (-0.466 - 0.884i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (-0.969 + 0.244i)T \)
53 \( 1 + (0.710 - 0.703i)T \)
59 \( 1 + (-0.999 + 0.0380i)T \)
61 \( 1 + (-0.0665 + 0.997i)T \)
67 \( 1 + (-0.928 + 0.371i)T \)
71 \( 1 + (0.516 - 0.856i)T \)
73 \( 1 + (0.625 - 0.780i)T \)
79 \( 1 + (0.398 - 0.917i)T \)
83 \( 1 + (-0.774 - 0.633i)T \)
89 \( 1 + (0.327 + 0.945i)T \)
97 \( 1 + (-0.736 - 0.676i)T \)
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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

−18.40806343351569583269921303418, −18.14019910436288712034419405823, −17.05812676054753636734173075230, −16.69350303401176542475457732317, −15.673467378325286488686248351, −15.111445516900011937068934046357, −14.37803958442155664833521393759, −13.399074503745177881354599680431, −12.97616836710268957293566301250, −12.41663117743424503355187938615, −11.32354882027645605203178649099, −11.02805184761277564959177282082, −10.006068286367271304039148565612, −9.31773723869906187700891615295, −8.802994771053448738258061515301, −8.23340201850178123680666365547, −7.47974589608655587718569067778, −6.7423507671643896267733594950, −6.2222003098669398140957354878, −4.706504843144279518336982821189, −3.92251534902451040139993394437, −3.258704876834420853650414907147, −2.48119478168066497541583264852, −1.65896093851641566957083144618, −1.12981730110560273038000862918, 0.472383746283091685745675233064, 1.600896014534338275163858696495, 2.3844095770246806196379668400, 3.17715103714008942056301060580, 4.11863427558680562116351355344, 5.01256035220887726115893377046, 5.71303910710171992458468560319, 6.62041208357866599541000404387, 7.445182338502482927629493542849, 7.92580130973025785944131043928, 8.7593238506938153675243871701, 9.10963201693575475085835059498, 9.95855336747880439229113815271, 10.5776800619874657989922088882, 11.0757107028336886733836873467, 12.159058491552962112698638891831, 13.139011225561649776636120169653, 13.71990048219783664924890183981, 14.52564409892553126544987621524, 15.031485954080689235368082174383, 15.5982370925651025395960012525, 16.3241587726797407853527843078, 16.75993521531583761576306259792, 17.75525726216001862216922500337, 18.32952812405047990048284101068

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