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Properties

Label 1-1872-1872.43-r1-0-0
Degree $1$
Conductor $1872$
Sign $-0.850 + 0.526i$
Analytic cond. $201.174$
Root an. cond. $201.174$
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  + (0.866 + 0.5i)5-s − 7-s + (−0.866 − 0.5i)11-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + 23-s + (0.5 + 0.866i)25-s + (0.866 + 0.5i)29-s + (−0.5 + 0.866i)31-s + (−0.866 − 0.5i)35-s + (−0.866 + 0.5i)37-s + 41-s − i·43-s + (−0.5 − 0.866i)47-s + 49-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)5-s − 7-s + (−0.866 − 0.5i)11-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + 23-s + (0.5 + 0.866i)25-s + (0.866 + 0.5i)29-s + (−0.5 + 0.866i)31-s + (−0.866 − 0.5i)35-s + (−0.866 + 0.5i)37-s + 41-s − i·43-s + (−0.5 − 0.866i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $-0.850 + 0.526i$
Analytic conductor: \(201.174\)
Root analytic conductor: \(201.174\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1872,\ (1:\ ),\ -0.850 + 0.526i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3084818385 + 1.084181374i\)
\(L(\frac12)\) \(\approx\) \(0.3084818385 + 1.084181374i\)
\(L(1)\) \(\approx\) \(0.9791038506 + 0.2020077878i\)
\(L(1)\) \(\approx\) \(0.9791038506 + 0.2020077878i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
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 \( 1 + T \)
43 \( 1 - iT \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + iT \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 - iT \)
67 \( 1 - iT \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 - T \)
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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

−19.63793921104051292827762675821, −18.927743906211334046880696741941, −17.8253820324494525347844691974, −17.74739326270635038943936181266, −16.51759804503509461611529537279, −16.07077383845560298558369835234, −15.37455026490870667435239101480, −14.34218530567467902451847564931, −13.44316536481454689823659857986, −13.100833322315549919286825646115, −12.40625112669388761108153930139, −11.40179524728821696186052060176, −10.50167970220899522943719077196, −9.58139208701577214988207994773, −9.41468511062684115644584634493, −8.37060355594704325877494969829, −7.28271256285267998313399926919, −6.68573998200541512719692644538, −5.675494099208819192968495844070, −5.09256876368897706540425240444, −4.19604210493905305086055527042, −2.83697963237924589060911696944, −2.48798384327724028839614996635, −1.13402999743108499708675522152, −0.216367623908084538941482772903, 1.06832038299345327862588720138, 2.18144828849131377697466435589, 3.06035062743526768802560329250, 3.60131926852236708052875248059, 5.07168838864304864485586912936, 5.64728307456446479409064922328, 6.55378752000631930380092379499, 7.04693912017428952591111037134, 8.198918733090703155426735421935, 9.0344751467223564953891857902, 9.77605033375249349955589928349, 10.55620911296018100541365710152, 10.93753151380615411672985867124, 12.2817113761926559024047975425, 12.85758258315007163392479744176, 13.63392898944829407020380357389, 14.11691015194721613236695854421, 15.177417062985178350740827542542, 15.793466643462070716611092143623, 16.591144405804651944996714318572, 17.29033463070196548293683585592, 18.15292702050600736416218417445, 18.67876617149202639795427408954, 19.41358679276978129638729742859, 20.17619780285744557029619528713

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