↑

Properties

Label 1-6003-6003.1199-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.507 + 0.861i$
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  + (−0.935 + 0.352i)2-s + (0.751 − 0.659i)4-s + (−0.675 − 0.737i)5-s + (−0.546 + 0.837i)7-s + (−0.470 + 0.882i)8-s + (0.891 + 0.452i)10-s + (0.0135 + 0.999i)11-s + (−0.314 + 0.949i)13-s + (0.215 − 0.976i)14-s + (0.128 − 0.991i)16-s + (0.540 − 0.841i)17-s + (−0.320 + 0.947i)19-s + (−0.994 − 0.108i)20-s + (−0.365 − 0.930i)22-s + (−0.0882 + 0.996i)25-s + (−0.0407 − 0.999i)26-s + ⋯
L(s)  = 1  + (−0.935 + 0.352i)2-s + (0.751 − 0.659i)4-s + (−0.675 − 0.737i)5-s + (−0.546 + 0.837i)7-s + (−0.470 + 0.882i)8-s + (0.891 + 0.452i)10-s + (0.0135 + 0.999i)11-s + (−0.314 + 0.949i)13-s + (0.215 − 0.976i)14-s + (0.128 − 0.991i)16-s + (0.540 − 0.841i)17-s + (−0.320 + 0.947i)19-s + (−0.994 − 0.108i)20-s + (−0.365 − 0.930i)22-s + (−0.0882 + 0.996i)25-s + (−0.0407 − 0.999i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.507 + 0.861i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.507 + 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3300636859 + 0.5772766831i\)
\(L(\frac12)\) \(\approx\) \(0.3300636859 + 0.5772766831i\)
\(L(1)\) \(\approx\) \(0.5515692968 + 0.1677993350i\)
\(L(1)\) \(\approx\) \(0.5515692968 + 0.1677993350i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.935 + 0.352i)T \)
5 \( 1 + (-0.675 - 0.737i)T \)
7 \( 1 + (-0.546 + 0.837i)T \)
11 \( 1 + (0.0135 + 0.999i)T \)
13 \( 1 + (-0.314 + 0.949i)T \)
17 \( 1 + (0.540 - 0.841i)T \)
19 \( 1 + (-0.320 + 0.947i)T \)
31 \( 1 + (-0.993 - 0.115i)T \)
37 \( 1 + (-0.0815 - 0.996i)T \)
41 \( 1 + (-0.371 + 0.928i)T \)
43 \( 1 + (0.993 - 0.115i)T \)
47 \( 1 + (0.294 + 0.955i)T \)
53 \( 1 + (0.768 - 0.639i)T \)
59 \( 1 + (0.995 + 0.0950i)T \)
61 \( 1 + (0.649 + 0.760i)T \)
67 \( 1 + (0.999 + 0.0135i)T \)
71 \( 1 + (-0.0611 - 0.998i)T \)
73 \( 1 + (0.852 + 0.523i)T \)
79 \( 1 + (0.384 - 0.923i)T \)
83 \( 1 + (-0.869 - 0.494i)T \)
89 \( 1 + (0.965 - 0.262i)T \)
97 \( 1 + (-0.229 + 0.973i)T \)
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

−17.381582370645885640495329543092, −17.02568735635335766737301376423, −16.28339283001608489837537199912, −15.63560220597786958520812591655, −15.123439402973177018282623709423, −14.27843416221971600677655559459, −13.43525420362643715155545314804, −12.74571676159335762619640280545, −12.14598975761637857996468893237, −11.16641096933019044429151135822, −10.89510370259333828886749924001, −10.26127218906911640713400009593, −9.72199094707851802518722403636, −8.64153489941465513845862822687, −8.23527747602908472902320565299, −7.41455268679073668426631659759, −6.97571172573522739090781865836, −6.26442927038183147178838917310, −5.40913997161898213215303889145, −4.04031239560793995489977877268, −3.571447367220593635939110197989, −2.9922000357569570105607648426, −2.226410469948161914593885298816, −0.93622213278892442596097011762, −0.36232462605395578265469792320, 0.80049104124789514656099560445, 1.84765446110789973326052002159, 2.353373886154197308558292805590, 3.46616609486007126498845583336, 4.34857604526110161477862897900, 5.20615105878592798728330871989, 5.74775998605015627408348860015, 6.69757135918225837846167676852, 7.29950958249017403920931928286, 7.8952704171328623082521587278, 8.6696299557171620458391429504, 9.39675206419799113778444807335, 9.547475920062246981291525645528, 10.48845718965698307894829570505, 11.43944570196829901576186797662, 11.99383609235051239598346514252, 12.39787375087292590551237590120, 13.17332927991735886883453415705, 14.50253726925476727587782681746, 14.658958411683220188528580726876, 15.584896774239153668853564033310, 16.16879012815685809651338320369, 16.45675217702641404270314158718, 17.20257614209828908481494036580, 17.97697584890096643217978521331

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