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Properties

Label 1-4235-4235.2707-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.300 - 0.953i$
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  + (0.371 + 0.928i)2-s + (−0.866 − 0.5i)3-s + (−0.723 + 0.690i)4-s + (0.142 − 0.989i)6-s + (−0.909 − 0.415i)8-s + (0.5 + 0.866i)9-s + (0.971 − 0.235i)12-s + (0.281 − 0.959i)13-s + (0.0475 − 0.998i)16-s + (−0.945 + 0.327i)17-s + (−0.618 + 0.786i)18-s + (−0.327 + 0.945i)19-s + (−0.998 − 0.0475i)23-s + (0.580 + 0.814i)24-s + (0.995 − 0.0950i)26-s − i·27-s + ⋯
L(s)  = 1  + (0.371 + 0.928i)2-s + (−0.866 − 0.5i)3-s + (−0.723 + 0.690i)4-s + (0.142 − 0.989i)6-s + (−0.909 − 0.415i)8-s + (0.5 + 0.866i)9-s + (0.971 − 0.235i)12-s + (0.281 − 0.959i)13-s + (0.0475 − 0.998i)16-s + (−0.945 + 0.327i)17-s + (−0.618 + 0.786i)18-s + (−0.327 + 0.945i)19-s + (−0.998 − 0.0475i)23-s + (0.580 + 0.814i)24-s + (0.995 − 0.0950i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.300 - 0.953i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.300 - 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2492031997 - 0.1826796698i\)
\(L(\frac12)\) \(\approx\) \(0.2492031997 - 0.1826796698i\)
\(L(1)\) \(\approx\) \(0.6785425298 + 0.2717862245i\)
\(L(1)\) \(\approx\) \(0.6785425298 + 0.2717862245i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.371 + 0.928i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (0.281 - 0.959i)T \)
17 \( 1 + (-0.945 + 0.327i)T \)
19 \( 1 + (-0.327 + 0.945i)T \)
23 \( 1 + (-0.998 - 0.0475i)T \)
29 \( 1 + (0.654 + 0.755i)T \)
31 \( 1 + (-0.235 + 0.971i)T \)
37 \( 1 + (-0.690 + 0.723i)T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (-0.909 - 0.415i)T \)
47 \( 1 + (0.618 + 0.786i)T \)
53 \( 1 + (0.998 - 0.0475i)T \)
59 \( 1 + (0.928 + 0.371i)T \)
61 \( 1 + (0.786 - 0.618i)T \)
67 \( 1 + (0.618 - 0.786i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (-0.458 - 0.888i)T \)
79 \( 1 + (-0.580 + 0.814i)T \)
83 \( 1 + (0.540 + 0.841i)T \)
89 \( 1 + (0.981 + 0.189i)T \)
97 \( 1 + (-0.909 - 0.415i)T \)
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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

−18.54414091758808373863232259156, −17.76825062913618136298215572589, −17.44369336020142404529729436427, −16.40765275748113139979528173174, −15.82024540612908802000372658475, −15.09744424416368439096957847145, −14.41702310597395001320563853279, −13.46637374118879457899187465612, −13.12213230047676394887796409212, −12.10712765414076489462526581802, −11.50046504919472816858532294194, −11.285360802020790900569970878924, −10.31481852575351201839626732928, −9.831775523144949491177459696107, −9.05643753755748271074404912740, −8.48599189455870492720079842004, −7.07497464776291000931635213536, −6.420828385387604172098275827491, −5.72315358707518742142578616834, −4.92051206656324759572596301952, −4.1994001177185767744854271843, −3.88417469434935521190252387487, −2.63102086952336960936655582008, −1.95425755544428525719034449987, −0.86002757487894241674840156691, 0.10848341906571776032258182629, 1.31030590729685264773516284350, 2.358578318625435702075404402578, 3.50559620119026914506348683604, 4.200529790092754020281636936460, 5.15422110747507735817558949581, 5.559504723542761264199645955824, 6.44175904542402657628780794945, 6.78917630491291741883706367739, 7.74592137382683562252186104671, 8.28191321124852702769866317662, 8.96589556990052050794112845313, 10.26416108897285604045814310156, 10.52275262786626541202970038164, 11.65844460824290849710967166375, 12.33951895356219996317287701227, 12.74244676782245949186656826575, 13.54611299364893792592536590296, 14.06120770952006451497878808541, 14.96662749628415375989368482189, 15.704899932628073458643393441987, 16.15255693862391907756964897960, 16.886146095766884982896439033057, 17.58455951204585437871192199237, 17.95876843054301060082199375194

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