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Properties

Label 1-4235-4235.557-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.594 - 0.804i$
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  + (0.875 − 0.483i)2-s + (0.994 + 0.104i)3-s + (0.532 − 0.846i)4-s + (0.921 − 0.389i)6-s + (0.0570 − 0.998i)8-s + (0.978 + 0.207i)9-s + (0.618 − 0.786i)12-s + (−0.170 − 0.985i)13-s + (−0.432 − 0.901i)16-s + (−0.803 + 0.595i)17-s + (0.956 − 0.290i)18-s + (0.00951 − 0.999i)19-s + (−0.690 − 0.723i)23-s + (0.161 − 0.986i)24-s + (−0.625 − 0.780i)26-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (0.875 − 0.483i)2-s + (0.994 + 0.104i)3-s + (0.532 − 0.846i)4-s + (0.921 − 0.389i)6-s + (0.0570 − 0.998i)8-s + (0.978 + 0.207i)9-s + (0.618 − 0.786i)12-s + (−0.170 − 0.985i)13-s + (−0.432 − 0.901i)16-s + (−0.803 + 0.595i)17-s + (0.956 − 0.290i)18-s + (0.00951 − 0.999i)19-s + (−0.690 − 0.723i)23-s + (0.161 − 0.986i)24-s + (−0.625 − 0.780i)26-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.594 - 0.804i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.594 - 0.804i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.775359153 - 3.520385921i\)
\(L(\frac12)\) \(\approx\) \(1.775359153 - 3.520385921i\)
\(L(1)\) \(\approx\) \(1.986269495 - 1.105310917i\)
\(L(1)\) \(\approx\) \(1.986269495 - 1.105310917i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.875 - 0.483i)T \)
3 \( 1 + (0.994 + 0.104i)T \)
13 \( 1 + (-0.170 - 0.985i)T \)
17 \( 1 + (-0.803 + 0.595i)T \)
19 \( 1 + (0.00951 - 0.999i)T \)
23 \( 1 + (-0.690 - 0.723i)T \)
29 \( 1 + (0.198 + 0.980i)T \)
31 \( 1 + (-0.830 - 0.556i)T \)
37 \( 1 + (-0.244 - 0.969i)T \)
41 \( 1 + (-0.0855 - 0.996i)T \)
43 \( 1 + (0.540 + 0.841i)T \)
47 \( 1 + (-0.956 - 0.290i)T \)
53 \( 1 + (0.901 + 0.432i)T \)
59 \( 1 + (0.905 - 0.424i)T \)
61 \( 1 + (0.999 - 0.0190i)T \)
67 \( 1 + (-0.945 - 0.327i)T \)
71 \( 1 + (-0.736 - 0.676i)T \)
73 \( 1 + (0.647 - 0.761i)T \)
79 \( 1 + (-0.710 - 0.703i)T \)
83 \( 1 + (-0.999 - 0.0285i)T \)
89 \( 1 + (0.995 + 0.0950i)T \)
97 \( 1 + (-0.633 + 0.774i)T \)
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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

−18.61978224059559358115327679963, −17.93780787771387542599113228178, −17.11602921469144857720945742736, −16.22130075362166122366000695075, −15.88729751109129972861211421813, −15.05784952556887932901556085541, −14.46613414924343685605905447134, −13.93160192536847002911659973311, −13.36876803582325112044737337101, −12.75400711802230267528460959048, −11.82468904065041168191243487451, −11.48731283075610600083834821277, −10.24423966258393819746083124638, −9.5741854492281896956211040819, −8.68628696859690880068121191635, −8.18671277842023604237919139156, −7.317488230357171541573768972, −6.86528076481660548106505360660, −6.055337421322111725165390367753, −5.13912784385298856960492424915, −4.24713681027792066612290057702, −3.85400811275617161994942623937, −2.940821760181638655752819918131, −2.18756369504293154160115890102, −1.52531470125593033149795992739, 0.591398332197649373158160186395, 1.80440026245663675628960111832, 2.364228357372085415016891834728, 3.116817262047305992526298182561, 3.82470063092935398046377594518, 4.5069188934128128285877433196, 5.24494064112759001093851240355, 6.12514018455996374938018696230, 6.98098340932493537024989917293, 7.58707034333379451261343266540, 8.59672459206744395002140078416, 9.14511602556063791942207536282, 10.071109130649276995765115911258, 10.594288549022847646103988192778, 11.22076436644612775795343718096, 12.28056722406328281960587476293, 12.88599593288458185334947395875, 13.260437993943130717359812167757, 14.08229367486690603753337897858, 14.71121417321164882035842791107, 15.17187224341385656821190173752, 15.857122090033996873002775890062, 16.43154918780554882964851198222, 17.74311934422744054531709340590, 18.18443942615384775655345621621

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