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Properties

Label 1-4235-4235.4163-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.999 + 0.0355i$
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  + (−0.962 − 0.272i)2-s + (−0.207 + 0.978i)3-s + (0.851 + 0.524i)4-s + (0.466 − 0.884i)6-s + (−0.676 − 0.736i)8-s + (−0.913 − 0.406i)9-s + (−0.690 + 0.723i)12-s + (−0.791 − 0.610i)13-s + (0.449 + 0.893i)16-s + (0.424 − 0.905i)17-s + (0.768 + 0.640i)18-s + (0.123 + 0.992i)19-s + (0.458 − 0.888i)23-s + (0.861 − 0.508i)24-s + (0.595 + 0.803i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.962 − 0.272i)2-s + (−0.207 + 0.978i)3-s + (0.851 + 0.524i)4-s + (0.466 − 0.884i)6-s + (−0.676 − 0.736i)8-s + (−0.913 − 0.406i)9-s + (−0.690 + 0.723i)12-s + (−0.791 − 0.610i)13-s + (0.449 + 0.893i)16-s + (0.424 − 0.905i)17-s + (0.768 + 0.640i)18-s + (0.123 + 0.992i)19-s + (0.458 − 0.888i)23-s + (0.861 − 0.508i)24-s + (0.595 + 0.803i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.999 + 0.0355i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (4163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.999 + 0.0355i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001096746525 + 0.06167321472i\)
\(L(\frac12)\) \(\approx\) \(0.001096746525 + 0.06167321472i\)
\(L(1)\) \(\approx\) \(0.5341868922 + 0.05841685748i\)
\(L(1)\) \(\approx\) \(0.5341868922 + 0.05841685748i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.962 - 0.272i)T \)
3 \( 1 + (-0.207 + 0.978i)T \)
13 \( 1 + (-0.791 - 0.610i)T \)
17 \( 1 + (0.424 - 0.905i)T \)
19 \( 1 + (0.123 + 0.992i)T \)
23 \( 1 + (0.458 - 0.888i)T \)
29 \( 1 + (-0.516 - 0.856i)T \)
31 \( 1 + (0.179 - 0.983i)T \)
37 \( 1 + (0.0760 - 0.997i)T \)
41 \( 1 + (-0.897 + 0.441i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 + (-0.768 + 0.640i)T \)
53 \( 1 + (0.893 + 0.449i)T \)
59 \( 1 + (-0.830 + 0.556i)T \)
61 \( 1 + (0.969 + 0.244i)T \)
67 \( 1 + (0.371 - 0.928i)T \)
71 \( 1 + (0.974 - 0.226i)T \)
73 \( 1 + (0.263 - 0.964i)T \)
79 \( 1 + (-0.749 + 0.662i)T \)
83 \( 1 + (-0.931 + 0.362i)T \)
89 \( 1 + (-0.327 + 0.945i)T \)
97 \( 1 + (0.491 - 0.870i)T \)
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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

−18.00629896479524252115152689302, −17.28216845723359461046468960077, −17.03296659189512003140260368931, −16.268215434820946696890399903377, −15.363714931683028676690822472680, −14.73279328322452156993398588588, −14.06016355820356170205956586860, −13.21848333384764596731387067085, −12.487744166472653271354561146428, −11.67935748671329050521011079952, −11.3397261099056857108350854809, −10.39133360331098701463588458088, −9.75246412590710856303869336995, −8.79930461863116820804970420375, −8.40357828952121972123487588199, −7.502092854505120414347212928170, −6.88928613805021223938323941144, −6.567827782763699558276229307240, −5.40048844291632783448396615520, −5.067648730616702291772009715732, −3.491233430349472673060592572310, −2.6705978484596769236305892459, −1.76840238590810058104800532058, −1.27287692287075520902990991097, −0.028981491922879016925895243335, 0.91310316321991736758449069154, 2.20866522857044168461839032140, 2.89210423686318767963599019081, 3.61889956066068307971215715103, 4.49720895787425872792128645110, 5.38980906026362417435170208821, 6.08238287992679691296074317160, 6.98922884650611648024400280215, 7.87157641001738652648955049095, 8.33588493268594301838086059211, 9.401246651869517059684752948657, 9.69645279627161488516669307808, 10.33741671456251625733540825646, 11.015296085927378006094648633481, 11.732337104511013939404886306851, 12.24215926438127027740153946402, 13.09198331184175647753516168591, 14.203381233892166095445039798337, 14.93423660489402474396796628241, 15.39442292540727065557017092818, 16.259048831471399170084045612500, 16.7836049570062403414139600635, 17.1315024790138717071899364139, 18.12990324243862711919034703566, 18.52096431373771597239604240068

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