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Properties

Label 1-4235-4235.1143-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.100 - 0.994i$
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  + (0.458 − 0.888i)2-s + (−0.866 − 0.5i)3-s + (−0.580 − 0.814i)4-s + (−0.841 + 0.540i)6-s + (−0.989 + 0.142i)8-s + (0.5 + 0.866i)9-s + (0.0950 + 0.995i)12-s + (−0.909 − 0.415i)13-s + (−0.327 + 0.945i)16-s + (0.690 + 0.723i)17-s + (0.998 − 0.0475i)18-s + (0.723 + 0.690i)19-s + (−0.945 − 0.327i)23-s + (0.928 + 0.371i)24-s + (−0.786 + 0.618i)26-s − i·27-s + ⋯
L(s)  = 1  + (0.458 − 0.888i)2-s + (−0.866 − 0.5i)3-s + (−0.580 − 0.814i)4-s + (−0.841 + 0.540i)6-s + (−0.989 + 0.142i)8-s + (0.5 + 0.866i)9-s + (0.0950 + 0.995i)12-s + (−0.909 − 0.415i)13-s + (−0.327 + 0.945i)16-s + (0.690 + 0.723i)17-s + (0.998 − 0.0475i)18-s + (0.723 + 0.690i)19-s + (−0.945 − 0.327i)23-s + (0.928 + 0.371i)24-s + (−0.786 + 0.618i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.100 - 0.994i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.100 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6402958240 - 0.7082850838i\)
\(L(\frac12)\) \(\approx\) \(0.6402958240 - 0.7082850838i\)
\(L(1)\) \(\approx\) \(0.6619365495 - 0.4772068834i\)
\(L(1)\) \(\approx\) \(0.6619365495 - 0.4772068834i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.458 - 0.888i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-0.909 - 0.415i)T \)
17 \( 1 + (0.690 + 0.723i)T \)
19 \( 1 + (0.723 + 0.690i)T \)
23 \( 1 + (-0.945 - 0.327i)T \)
29 \( 1 + (-0.959 + 0.281i)T \)
31 \( 1 + (-0.995 - 0.0950i)T \)
37 \( 1 + (-0.814 - 0.580i)T \)
41 \( 1 + (-0.841 + 0.540i)T \)
43 \( 1 + (-0.989 + 0.142i)T \)
47 \( 1 + (-0.998 - 0.0475i)T \)
53 \( 1 + (0.945 - 0.327i)T \)
59 \( 1 + (0.888 - 0.458i)T \)
61 \( 1 + (-0.0475 + 0.998i)T \)
67 \( 1 + (0.998 - 0.0475i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (-0.189 - 0.981i)T \)
79 \( 1 + (0.928 - 0.371i)T \)
83 \( 1 + (0.755 + 0.654i)T \)
89 \( 1 + (-0.235 - 0.971i)T \)
97 \( 1 + (-0.989 + 0.142i)T \)
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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

−18.25431914742546331156181088629, −17.70958367645553027347709052599, −16.95947559428099499371615650006, −16.5297016669527456381156502602, −15.92171099815413802590137908062, −15.246638772743082318269316798866, −14.65433959866440634988225957250, −13.902746087391983857123187450843, −13.21890415332084236999352330734, −12.308755475025063568479996796431, −11.83972192024231036994731883037, −11.30101805627233912453857580917, −10.08016851586422167009677274137, −9.63361353287455442406298113907, −8.950149552261512514767414627175, −7.92734543734980569315250531317, −7.1063765421496164185575579019, −6.7900626056993707798471259575, −5.67713889008477595169098007200, −5.2740009454180949691472385344, −4.684722923891575988823469097162, −3.77140063745642525492835313919, −3.19252849265864725498782742170, −1.928185925551345899680190167724, −0.49089512161051816803327182799, 0.50501346564366714016268448064, 1.65263259288226987686588334901, 2.0198562762735187290806467451, 3.24028023588897692373075149223, 3.87577642268095907416447699305, 4.898606227695500081584307501495, 5.45305572285104971038253181513, 5.96933753098072233782165997993, 6.91674279583176826467096873300, 7.717932593084402108866343105309, 8.47397238036470250683263009925, 9.6328016148835193093248662718, 10.16759158474701957021929069116, 10.65067416429477992925764020072, 11.68376719138786653688707407096, 11.90173592611015933347304313917, 12.76453644555010115409542597559, 13.05102002666216623903626280300, 14.04486359214061758869458457015, 14.61758803958227395983668610717, 15.316404773516360608348537585521, 16.437714392525963379116013223661, 16.78626806529057281165989804957, 17.93059402142202150629906974009, 18.07427716170956958530621893681

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