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Properties

Label 1-4235-4235.1068-r1-0-0
Degree $1$
Conductor $4235$
Sign $0.733 - 0.679i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (−0.945 + 0.327i)2-s + (0.866 − 0.5i)3-s + (0.786 − 0.618i)4-s + (−0.654 + 0.755i)6-s + (−0.540 + 0.841i)8-s + (0.5 − 0.866i)9-s + (0.371 − 0.928i)12-s + (0.989 − 0.142i)13-s + (0.235 − 0.971i)16-s + (0.0950 + 0.995i)17-s + (−0.189 + 0.981i)18-s + (0.995 + 0.0950i)19-s + (0.971 + 0.235i)23-s + (−0.0475 + 0.998i)24-s + (−0.888 + 0.458i)26-s − i·27-s + ⋯
L(s)  = 1  + (−0.945 + 0.327i)2-s + (0.866 − 0.5i)3-s + (0.786 − 0.618i)4-s + (−0.654 + 0.755i)6-s + (−0.540 + 0.841i)8-s + (0.5 − 0.866i)9-s + (0.371 − 0.928i)12-s + (0.989 − 0.142i)13-s + (0.235 − 0.971i)16-s + (0.0950 + 0.995i)17-s + (−0.189 + 0.981i)18-s + (0.995 + 0.0950i)19-s + (0.971 + 0.235i)23-s + (−0.0475 + 0.998i)24-s + (−0.888 + 0.458i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.733 - 0.679i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1068, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ 0.733 - 0.679i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.449878272 - 0.9597668862i\)
\(L(\frac12)\) \(\approx\) \(2.449878272 - 0.9597668862i\)
\(L(1)\) \(\approx\) \(1.112016293 - 0.1190980475i\)
\(L(1)\) \(\approx\) \(1.112016293 - 0.1190980475i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.945 + 0.327i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (0.989 - 0.142i)T \)
17 \( 1 + (0.0950 + 0.995i)T \)
19 \( 1 + (0.995 + 0.0950i)T \)
23 \( 1 + (0.971 + 0.235i)T \)
29 \( 1 + (-0.415 - 0.909i)T \)
31 \( 1 + (0.928 - 0.371i)T \)
37 \( 1 + (-0.618 + 0.786i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (-0.540 + 0.841i)T \)
47 \( 1 + (-0.189 - 0.981i)T \)
53 \( 1 + (-0.971 + 0.235i)T \)
59 \( 1 + (0.327 - 0.945i)T \)
61 \( 1 + (0.981 - 0.189i)T \)
67 \( 1 + (0.189 - 0.981i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (-0.690 + 0.723i)T \)
79 \( 1 + (-0.0475 - 0.998i)T \)
83 \( 1 + (0.281 - 0.959i)T \)
89 \( 1 + (-0.580 - 0.814i)T \)
97 \( 1 + (0.540 - 0.841i)T \)
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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

−18.38922037904423870212929519888, −17.79389405707366965102334070358, −16.86804836558120409060678746045, −16.13634790589659704488801314118, −15.8262502140281437973996068685, −15.10741241054341682303285501720, −14.1768393419131663229651364308, −13.60457413761456464621427081744, −12.83547076913581895756809184364, −11.97494265972844371527131459668, −11.19610653361219387560187150160, −10.64659852635690223446636652355, −9.90800732588663437236393909265, −9.21906408746098197506261077508, −8.77050706187136336746541771479, −8.122845159466383110678465672, −7.22209026423976353647383093201, −6.84932858993608856109659962228, −5.57544064407198926351426707558, −4.75622136330816492662156332786, −3.65854067995175648071129987036, −3.21464145124468238134094338257, −2.45833809633032514173933567241, −1.51921639607131502890585466676, −0.78156250378151849861232493957, 0.57923927649464115714063151365, 1.357991755351903204202756022810, 1.92386213378405654489818141497, 3.04114191410794301225269950411, 3.50084766180078162078623665327, 4.74145615032338571359242685208, 5.80967472410215025534309597272, 6.43566979339366121112731743931, 7.07261639260531130016862602449, 7.99413549169497591780599527069, 8.26447114920260675185046903154, 8.996564125341207463361974272273, 9.78484356218951531885765529383, 10.23652480781778534301922839429, 11.34761403162118835973603811456, 11.73076234686931366686514926691, 12.8539821120592765663902089934, 13.39721103432645957093971183534, 14.16829776738151480242323822875, 14.89210272435340537700396760737, 15.47328222350813748657756735601, 15.961501800178661584495568142702, 17.024133840765538446881455238901, 17.42308834683543686203575446553, 18.3683890010834731955254293783

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