↑

Properties

Label 1-6003-6003.1076-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.635 + 0.772i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.635 + 0.772i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1076, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.635 + 0.772i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.425076217 + 1.145067416i\)
\(L(\frac12)\) \(\approx\) \(2.425076217 + 1.145067416i\)
\(L(1)\) \(\approx\) \(1.499969688 + 0.4331604849i\)
\(L(1)\) \(\approx\) \(1.499969688 + 0.4331604849i\)

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

−17.92420599145966464407953057461, −16.77919523575986051547079417483, −16.167951274709384670283275322680, −15.41663723944444905795025122631, −14.79386127198655925253877514107, −14.375702097196475831048515831820, −13.905172082646092081490141079894, −12.90821401404814484885516646694, −12.03959917748427213725360717724, −11.85323568520659283708668487729, −11.249647408734027627765360505992, −10.510579598521752794549319943589, −9.8100205915194446639123971725, −9.151687964733528349397684874869, −8.05170163301493958017460791712, −7.63456860993902185545578490043, −6.6213205969383601490366703460, −5.938605617595369037713618867497, −5.40170144477409453297611577195, −4.527100055519207521598109190460, −3.85482828356215848173065242476, −2.96709243802809163847381556091, −2.65901219018241839996240030438, −1.67756152426353730423243821777, −0.67330986004841105697867090082, 0.776975793405776716728444935545, 1.75275303475470489511969909165, 2.748185243457262049563376297024, 3.63796574182501573139794335094, 4.33836787970592826459568748177, 4.83438965968790272175079224439, 5.26346430172094551926515688452, 6.40894714502999079332747982038, 7.11764709126217144819104897376, 7.74658199210399020584859329632, 8.00984151911314218121072602953, 9.05857394680512220694842528471, 9.71902435918856833542714613504, 10.68911984779018414031903302038, 11.43530143060987065707803405651, 12.20431911817173400065197619238, 12.431009334550977394777535413550, 13.27536936029179176812590021558, 13.899109429331350122028572950045, 14.67775814019314612674864319566, 15.08550536733615030706807016181, 15.72883213387524492786774508996, 16.65725564579007487365407615037, 16.88505152159821238591168885001, 17.55602696970010507432433522869

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