↑

Properties

Label 1-72e2-5184.995-r0-0-0
Degree $1$
Conductor $5184$
Sign $0.981 - 0.193i$
Analytic cond. $24.0743$
Root an. cond. $24.0743$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.696 + 0.717i)5-s + (−0.664 − 0.747i)7-s + (−0.858 − 0.512i)11-s + (0.934 − 0.355i)13-s + (0.984 − 0.173i)17-s + (0.843 + 0.537i)19-s + (−0.664 + 0.747i)23-s + (−0.0290 − 0.999i)25-s + (0.810 + 0.585i)29-s + (0.893 + 0.448i)31-s + (0.999 + 0.0436i)35-s + (−0.0436 − 0.999i)37-s + (−0.880 + 0.474i)41-s + (0.873 − 0.487i)43-s + (0.448 + 0.893i)47-s + ⋯
L(s)  = 1  + (−0.696 + 0.717i)5-s + (−0.664 − 0.747i)7-s + (−0.858 − 0.512i)11-s + (0.934 − 0.355i)13-s + (0.984 − 0.173i)17-s + (0.843 + 0.537i)19-s + (−0.664 + 0.747i)23-s + (−0.0290 − 0.999i)25-s + (0.810 + 0.585i)29-s + (0.893 + 0.448i)31-s + (0.999 + 0.0436i)35-s + (−0.0436 − 0.999i)37-s + (−0.880 + 0.474i)41-s + (0.873 − 0.487i)43-s + (0.448 + 0.893i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $0.981 - 0.193i$
Analytic conductor: \(24.0743\)
Root analytic conductor: \(24.0743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (995, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5184,\ (0:\ ),\ 0.981 - 0.193i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.273242862 - 0.1246244307i\)
\(L(\frac12)\) \(\approx\) \(1.273242862 - 0.1246244307i\)
\(L(1)\) \(\approx\) \(0.9014063538 + 0.002729995447i\)
\(L(1)\) \(\approx\) \(0.9014063538 + 0.002729995447i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.696 + 0.717i)T \)
7 \( 1 + (-0.664 - 0.747i)T \)
11 \( 1 + (-0.858 - 0.512i)T \)
13 \( 1 + (0.934 - 0.355i)T \)
17 \( 1 + (0.984 - 0.173i)T \)
19 \( 1 + (0.843 + 0.537i)T \)
23 \( 1 + (-0.664 + 0.747i)T \)
29 \( 1 + (0.810 + 0.585i)T \)
31 \( 1 + (0.893 + 0.448i)T \)
37 \( 1 + (-0.0436 - 0.999i)T \)
41 \( 1 + (-0.880 + 0.474i)T \)
43 \( 1 + (0.873 - 0.487i)T \)
47 \( 1 + (0.448 + 0.893i)T \)
53 \( 1 + (-0.793 - 0.608i)T \)
59 \( 1 + (-0.969 - 0.244i)T \)
61 \( 1 + (-0.561 - 0.827i)T \)
67 \( 1 + (-0.987 - 0.159i)T \)
71 \( 1 + (0.0871 - 0.996i)T \)
73 \( 1 + (0.0871 + 0.996i)T \)
79 \( 1 + (0.957 - 0.286i)T \)
83 \( 1 + (-0.994 - 0.101i)T \)
89 \( 1 + (0.996 - 0.0871i)T \)
97 \( 1 + (-0.396 + 0.918i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.214786554704028594025834235295, −17.25850774455836250824899278608, −16.504621090901710551040005389625, −15.93082664394135012055932899466, −15.532441952306948530444146647231, −14.957485240355893124695399362959, −13.75794542358400952378864249470, −13.41577597147968205863588374714, −12.44455237785609248619544546868, −12.13299400696639096750961720442, −11.53748791097986400071860898705, −10.52600717690223259307838811638, −9.88121746095579677319143980031, −9.18411356512422731421768898874, −8.42508086992306618128366118608, −7.975967093792261631075851105613, −7.1561622470875109163749825796, −6.22965724107234573759249018170, −5.64091171640959189047003447684, −4.80719434351543104973073467440, −4.194713216727780300113422923949, −3.229679641692049699782826302120, −2.67251757539581780858502841815, −1.568797144023081661233947208007, −0.64094390675805648939689449623, 0.565165764805183645793225440358, 1.426731948105441446488647678482, 2.83591331786060967823525561110, 3.30511189102914597900088877607, 3.752559067575704900698304386697, 4.77474068840835878953445446870, 5.73097396155812550628710353367, 6.28289235478948257051475278075, 7.13401903460203887221786103500, 7.86059516021589876132952751427, 8.1051224448736521393978465372, 9.28419500169182936740253459031, 10.069624127125059299729009845280, 10.61298200405055768661975311628, 11.06452906653283728437710105028, 12.044265272654176170214681591312, 12.48751661125811481233512053108, 13.557296987135853063087042221483, 13.873576517352167379443813492757, 14.530984787727825105723433909719, 15.633344373128378572803544464912, 15.93818069639468819808412390624, 16.3305819211309806568738945765, 17.38274530719091195134293591475, 18.08915399192158925831374742551

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