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Properties

Label 1-555-555.209-r0-0-0
Degree $1$
Conductor $555$
Sign $0.781 + 0.624i$
Analytic cond. $2.57740$
Root an. cond. $2.57740$
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  + (0.342 − 0.939i)2-s + (−0.766 − 0.642i)4-s + (−0.173 + 0.984i)7-s + (−0.866 + 0.5i)8-s + (−0.5 − 0.866i)11-s + (−0.642 + 0.766i)13-s + (0.866 + 0.5i)14-s + (0.173 + 0.984i)16-s + (−0.642 − 0.766i)17-s + (0.342 + 0.939i)19-s + (−0.984 + 0.173i)22-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)26-s + (0.766 − 0.642i)28-s + (−0.866 + 0.5i)29-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (−0.766 − 0.642i)4-s + (−0.173 + 0.984i)7-s + (−0.866 + 0.5i)8-s + (−0.5 − 0.866i)11-s + (−0.642 + 0.766i)13-s + (0.866 + 0.5i)14-s + (0.173 + 0.984i)16-s + (−0.642 − 0.766i)17-s + (0.342 + 0.939i)19-s + (−0.984 + 0.173i)22-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)26-s + (0.766 − 0.642i)28-s + (−0.866 + 0.5i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.781 + 0.624i$
Analytic conductor: \(2.57740\)
Root analytic conductor: \(2.57740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 555,\ (0:\ ),\ 0.781 + 0.624i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8036696642 + 0.2817596849i\)
\(L(\frac12)\) \(\approx\) \(0.8036696642 + 0.2817596849i\)
\(L(1)\) \(\approx\) \(0.8925895191 - 0.2280219401i\)
\(L(1)\) \(\approx\) \(0.8925895191 - 0.2280219401i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.342 - 0.939i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.642 + 0.766i)T \)
17 \( 1 + (-0.642 - 0.766i)T \)
19 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + iT \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (-0.984 + 0.173i)T \)
61 \( 1 + (-0.642 + 0.766i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.984 - 0.173i)T \)
83 \( 1 + (0.766 - 0.642i)T \)
89 \( 1 + (0.984 - 0.173i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−23.20109381082578496463329277643, −22.66203502234327607141423476617, −21.88131523336666483590317380402, −20.73414053988854377602016107713, −20.052581034119330978292851357674, −18.96320132334856302541714384947, −17.79910565121456792993039027753, −17.30449292947610590394316708470, −16.58193845873834944383884279352, −15.33527592912691499319378755439, −15.08935752253998448174850233648, −13.89155969065449117938135853267, −13.05980252869225149998460470919, −12.60065795203273156982962933295, −11.16129380168041254735481475743, −10.12890247355546006722333316527, −9.28135374850146663103153070875, −8.08067271399129623412825723654, −7.322791979662293582815483186372, −6.67160343293937775839764411679, −5.408161429111709146473771221332, −4.59351565838715571112724266614, −3.68954169511578318378973487639, −2.41444113354429741079871170360, −0.39822602359565777863309410519, 1.44847849828878131677464217997, 2.61456214091024761041835333228, 3.306314249830146962645076050197, 4.67859192273463363478172593627, 5.44069220301939385041414831336, 6.38738807613750159216101817520, 7.8342326084770147959041970589, 9.08276786083478486828526334057, 9.4139657777352201555444428763, 10.721507419661485677813808137215, 11.455030981711465695858397311587, 12.23769624198972362331628097996, 13.04576545900650830016848292518, 13.96861973747867882999008670134, 14.73376986751074769130795588032, 15.72903779458369684696647731541, 16.64268183859680038191452920117, 17.96390077468481943024180131228, 18.58312129167495065447981448613, 19.25140184822475255871740074763, 20.074731609879211743908963028575, 21.27158169937719069631423197992, 21.48278797315664968624658216424, 22.47597373911034106094206512327, 23.15797958035592725343686423509

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