↑

Properties

Label 1-6003-6003.106-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.0118 - 0.999i$
Analytic cond. $27.8778$
Root an. cond. $27.8778$
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  + (0.798 − 0.601i)2-s + (0.275 − 0.961i)4-s + (−0.634 + 0.773i)5-s + (−0.534 − 0.844i)7-s + (−0.359 − 0.933i)8-s + (−0.0407 + 0.999i)10-s + (−0.268 + 0.963i)11-s + (−0.994 + 0.108i)13-s + (−0.935 − 0.352i)14-s + (−0.848 − 0.529i)16-s + (−0.909 + 0.415i)17-s + (0.242 + 0.970i)19-s + (0.568 + 0.822i)20-s + (0.365 + 0.930i)22-s + (−0.195 − 0.980i)25-s + (−0.728 + 0.685i)26-s + ⋯
L(s)  = 1  + (0.798 − 0.601i)2-s + (0.275 − 0.961i)4-s + (−0.634 + 0.773i)5-s + (−0.534 − 0.844i)7-s + (−0.359 − 0.933i)8-s + (−0.0407 + 0.999i)10-s + (−0.268 + 0.963i)11-s + (−0.994 + 0.108i)13-s + (−0.935 − 0.352i)14-s + (−0.848 − 0.529i)16-s + (−0.909 + 0.415i)17-s + (0.242 + 0.970i)19-s + (0.568 + 0.822i)20-s + (0.365 + 0.930i)22-s + (−0.195 − 0.980i)25-s + (−0.728 + 0.685i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0118 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0118 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.0118 - 0.999i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.0118 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9077536148 - 0.9185793052i\)
\(L(\frac12)\) \(\approx\) \(0.9077536148 - 0.9185793052i\)
\(L(1)\) \(\approx\) \(1.049795834 - 0.3556276170i\)
\(L(1)\) \(\approx\) \(1.049795834 - 0.3556276170i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.798 - 0.601i)T \)
5 \( 1 + (-0.634 + 0.773i)T \)
7 \( 1 + (-0.534 - 0.844i)T \)
11 \( 1 + (-0.268 + 0.963i)T \)
13 \( 1 + (-0.994 + 0.108i)T \)
17 \( 1 + (-0.909 + 0.415i)T \)
19 \( 1 + (0.242 + 0.970i)T \)
31 \( 1 + (-0.737 + 0.675i)T \)
37 \( 1 + (0.998 + 0.0611i)T \)
41 \( 1 + (0.971 + 0.235i)T \)
43 \( 1 + (-0.737 - 0.675i)T \)
47 \( 1 + (0.294 + 0.955i)T \)
53 \( 1 + (-0.262 - 0.965i)T \)
59 \( 1 + (-0.327 - 0.945i)T \)
61 \( 1 + (-0.999 - 0.00679i)T \)
67 \( 1 + (0.963 - 0.268i)T \)
71 \( 1 + (-0.339 - 0.940i)T \)
73 \( 1 + (-0.999 + 0.0203i)T \)
79 \( 1 + (0.999 - 0.0339i)T \)
83 \( 1 + (0.612 - 0.790i)T \)
89 \( 1 + (0.830 + 0.557i)T \)
97 \( 1 + (0.996 + 0.0882i)T \)
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

−17.78464765871948755760427559122, −16.85213301401419077130779414457, −16.482392238860648245055119384108, −15.79929625842724783876846571389, −15.35033423950501696579377315289, −14.81748764763413967319243569233, −13.85787900816982658751814596818, −13.16145127533574500606274384529, −12.82732510601255398628270485916, −12.05572803567698307451457977828, −11.50265312538226136407202059575, −10.97643894882432699650902980172, −9.61997742702918889935604594922, −8.98307604941108347647287643322, −8.552422310923504724843004351787, −7.64376370372896582173003288388, −7.18924763004762633142141487031, −6.20530097329088648747890717447, −5.65542507022425196791888762467, −4.92031643167424893046062253921, −4.4284983824262390975355480811, −3.494552813426150350847417802553, −2.77763400292624417423770640071, −2.20744865888979790371753419095, −0.61931955882444579993599487468, 0.35865336407047689262146945793, 1.66672410766281290217532617862, 2.34318763640580410465519677552, 3.16585510906201139776692676306, 3.78177274776983105692659433507, 4.44483174006106257953918878254, 4.99544975662348687953089666949, 6.17423873027007140660404706576, 6.63708153097576865830867086797, 7.4044184720904962836061913068, 7.80910956350457459128693796471, 9.21513303631673558992204909317, 9.8347502305084696829422297518, 10.44621077684883848193018374833, 10.87973467390782873843514772418, 11.679370046203003480111186764471, 12.3935458722201411522593680748, 12.80882628973625303347759821326, 13.57434714202556337064349830607, 14.46468407129906168799289392673, 14.63936231296840200868027624363, 15.45985011088248275941110874109, 16.02134784759439509855286944238, 16.78547808797439709256265709073, 17.72570811759495914552223642963

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