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Properties

Label 1-2128-2128.331-r0-0-0
Degree $1$
Conductor $2128$
Sign $0.685 + 0.728i$
Analytic cond. $9.88239$
Root an. cond. $9.88239$
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  − i·3-s + (−0.866 − 0.5i)5-s − 9-s + (0.866 + 0.5i)11-s + (−0.866 − 0.5i)13-s + (−0.5 + 0.866i)15-s + 17-s + 23-s + (0.5 + 0.866i)25-s − i·27-s + (−0.866 − 0.5i)29-s + (−0.5 + 0.866i)31-s + (0.5 − 0.866i)33-s + (−0.866 + 0.5i)37-s + (−0.5 + 0.866i)39-s + ⋯
L(s)  = 1  − i·3-s + (−0.866 − 0.5i)5-s − 9-s + (0.866 + 0.5i)11-s + (−0.866 − 0.5i)13-s + (−0.5 + 0.866i)15-s + 17-s + 23-s + (0.5 + 0.866i)25-s − i·27-s + (−0.866 − 0.5i)29-s + (−0.5 + 0.866i)31-s + (0.5 − 0.866i)33-s + (−0.866 + 0.5i)37-s + (−0.5 + 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2128\)    =    \(2^{4} \cdot 7 \cdot 19\)
Sign: $0.685 + 0.728i$
Analytic conductor: \(9.88239\)
Root analytic conductor: \(9.88239\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2128} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2128,\ (0:\ ),\ 0.685 + 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5682721609 + 0.2455598304i\)
\(L(\frac12)\) \(\approx\) \(0.5682721609 + 0.2455598304i\)
\(L(1)\) \(\approx\) \(0.7533491547 - 0.2287676751i\)
\(L(1)\) \(\approx\) \(0.7533491547 - 0.2287676751i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 - iT \)
5 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 - iT \)
61 \( 1 - iT \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + iT \)
89 \( 1 + T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−19.64028468564054604487088933958, −19.173390217798075875537316382527, −18.449236119932457132693953255602, −17.24888100693182723093355267332, −16.69566877050393161436725609116, −16.20868590490814225140194863456, −15.25436138608947531436136852981, −14.546728601278560939241550653703, −14.44602775103842192849624065830, −13.17695838903177448205315916619, −12.0391917706258682043811468765, −11.59530486337724732345005389164, −10.93768722294824081462706901387, −10.121431670029135309818716074871, −9.356862249582859394865186043514, −8.69760686979375099452742403570, −7.76673737956554355330015534871, −7.00744399264898861672655014660, −6.10210964012660041348294522833, −5.1093192041468720606138666166, −4.42889809458081534640953351320, −3.40260463005494503784276364137, −3.22340324245453832531189708710, −1.79733705260254089120493149148, −0.23743091319563474736225515508, 1.0251240017176040697484389173, 1.74332386185236004338732557047, 2.988882071199405375117858599672, 3.658569654832872131133607876922, 4.86018587562282579155586071240, 5.42724355885278484592748827523, 6.59804049454765645523705790619, 7.27342877030469464513611200552, 7.798064056012893147412637377898, 8.64197810314012735090495548252, 9.339316473449602149878790863, 10.35356272465568916513235690162, 11.45174511920287238527672720494, 11.9162911598963894821464329186, 12.58917003437283189304325177087, 13.05251573318184477808275083586, 14.17162814647988215849148857825, 14.76335449421105832720454593202, 15.41056869292526600688728118898, 16.565045277677785728251788786188, 17.04288084431004857329951759758, 17.63593527545030246469831270252, 18.71184126118721915067265927945, 19.147091835771208152386906564043, 19.8633610886086345782766678536

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