Properties

Label 1-6003-6003.4-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.846 - 0.531i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.115 − 0.993i)2-s + (−0.973 − 0.229i)4-s + (−0.876 + 0.482i)5-s + (−0.612 − 0.790i)7-s + (−0.339 + 0.940i)8-s + (0.377 + 0.925i)10-s + (0.0339 − 0.999i)11-s + (−0.999 + 0.0135i)13-s + (−0.855 + 0.517i)14-s + (0.894 + 0.446i)16-s + (0.142 − 0.989i)17-s + (−0.685 + 0.728i)19-s + (0.963 − 0.268i)20-s + (−0.988 − 0.149i)22-s + (0.534 − 0.844i)25-s + (−0.101 + 0.994i)26-s + ⋯
L(s)  = 1  + (0.115 − 0.993i)2-s + (−0.973 − 0.229i)4-s + (−0.876 + 0.482i)5-s + (−0.612 − 0.790i)7-s + (−0.339 + 0.940i)8-s + (0.377 + 0.925i)10-s + (0.0339 − 0.999i)11-s + (−0.999 + 0.0135i)13-s + (−0.855 + 0.517i)14-s + (0.894 + 0.446i)16-s + (0.142 − 0.989i)17-s + (−0.685 + 0.728i)19-s + (0.963 − 0.268i)20-s + (−0.988 − 0.149i)22-s + (0.534 − 0.844i)25-s + (−0.101 + 0.994i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.846 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.846 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.846 - 0.531i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.846 - 0.531i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2305586987 - 0.8006791052i\)
\(L(\frac12)\) \(\approx\) \(0.2305586987 - 0.8006791052i\)
\(L(1)\) \(\approx\) \(0.5957711635 - 0.4229760633i\)
\(L(1)\) \(\approx\) \(0.5957711635 - 0.4229760633i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.115 - 0.993i)T \)
5 \( 1 + (-0.876 + 0.482i)T \)
7 \( 1 + (-0.612 - 0.790i)T \)
11 \( 1 + (0.0339 - 0.999i)T \)
13 \( 1 + (-0.999 + 0.0135i)T \)
17 \( 1 + (0.142 - 0.989i)T \)
19 \( 1 + (-0.685 + 0.728i)T \)
31 \( 1 + (0.476 - 0.879i)T \)
37 \( 1 + (0.979 + 0.202i)T \)
41 \( 1 + (-0.580 + 0.814i)T \)
43 \( 1 + (0.476 + 0.879i)T \)
47 \( 1 + (0.733 + 0.680i)T \)
53 \( 1 + (0.986 + 0.162i)T \)
59 \( 1 + (0.235 + 0.971i)T \)
61 \( 1 + (0.195 - 0.980i)T \)
67 \( 1 + (-0.0339 - 0.999i)T \)
71 \( 1 + (-0.591 - 0.806i)T \)
73 \( 1 + (-0.557 - 0.830i)T \)
79 \( 1 + (0.833 + 0.552i)T \)
83 \( 1 + (-0.275 + 0.961i)T \)
89 \( 1 + (0.992 - 0.122i)T \)
97 \( 1 + (0.546 + 0.837i)T \)
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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

−17.685404710078866070223118869996, −17.26234457271696836824156598142, −16.62702459302947158731420110552, −15.86426997425836310872248275724, −15.38367890204794708153903837030, −14.9294856621301222149050659304, −14.40049869301336834589165829045, −13.156389189102673016106213426589, −12.859756181170028356800659390561, −12.16213202265474205597818953447, −11.80987355748360898078583053937, −10.44170525662150294605732069231, −9.91218177221389480603010925012, −8.91733594149239608250072512875, −8.75387566488857212263547576164, −7.85765356070888029140221110413, −7.12662839296587555038756694256, −6.73241741539286356758369104813, −5.71605929547211788511317222707, −5.14662592667813124141932136810, −4.386249837030279650774344321964, −3.8872307059781353346368201915, −2.89448180181480840392039570948, −2.007844720270025639854156107240, −0.6394913880285472066242650421, 0.378061332895984212025774294870, 1.02344134400475626709390281804, 2.391113603798904226327061256587, 2.93763275945155339062524475067, 3.58344793323171567204414509837, 4.2672977644992457525722268091, 4.83235548157969142156086672482, 5.947709622374510281545693712089, 6.56854025036435169444554797114, 7.66988324464735551317632466028, 7.90478142304206653754564569199, 8.9634462237938358586621402845, 9.653392867922150065368110195260, 10.28107958952735559230265361593, 10.86982586736484606766339550931, 11.51637510280123341731526525176, 12.04438715021085371873041489823, 12.75130209452927199934223247168, 13.473698033821122079472312915406, 14.047478469951527331035634416223, 14.68590597696605978923085869529, 15.30972796697760896801256053039, 16.424610714307927471390268997870, 16.640847593936124096820440283273, 17.53672738713587703955185055540

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