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Properties

Label 1-6003-6003.1208-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.994 - 0.107i$
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  + (−0.346 + 0.938i)2-s + (−0.760 − 0.649i)4-s + (0.390 − 0.920i)5-s + (0.288 − 0.957i)7-s + (0.872 − 0.488i)8-s + (0.728 + 0.685i)10-s + (−0.746 + 0.665i)11-s + (0.568 − 0.822i)13-s + (0.798 + 0.601i)14-s + (0.155 + 0.987i)16-s + (−0.755 − 0.654i)17-s + (−0.983 − 0.182i)19-s + (−0.894 + 0.446i)20-s + (−0.365 − 0.930i)22-s + (−0.694 − 0.719i)25-s + (0.574 + 0.818i)26-s + ⋯
L(s)  = 1  + (−0.346 + 0.938i)2-s + (−0.760 − 0.649i)4-s + (0.390 − 0.920i)5-s + (0.288 − 0.957i)7-s + (0.872 − 0.488i)8-s + (0.728 + 0.685i)10-s + (−0.746 + 0.665i)11-s + (0.568 − 0.822i)13-s + (0.798 + 0.601i)14-s + (0.155 + 0.987i)16-s + (−0.755 − 0.654i)17-s + (−0.983 − 0.182i)19-s + (−0.894 + 0.446i)20-s + (−0.365 − 0.930i)22-s + (−0.694 − 0.719i)25-s + (0.574 + 0.818i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.994 - 0.107i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.994 - 0.107i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01636427874 - 0.3031020731i\)
\(L(\frac12)\) \(\approx\) \(0.01636427874 - 0.3031020731i\)
\(L(1)\) \(\approx\) \(0.7492034705 + 0.006619369185i\)
\(L(1)\) \(\approx\) \(0.7492034705 + 0.006619369185i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.346 + 0.938i)T \)
5 \( 1 + (0.390 - 0.920i)T \)
7 \( 1 + (0.288 - 0.957i)T \)
11 \( 1 + (-0.746 + 0.665i)T \)
13 \( 1 + (0.568 - 0.822i)T \)
17 \( 1 + (-0.755 - 0.654i)T \)
19 \( 1 + (-0.983 - 0.182i)T \)
31 \( 1 + (0.773 + 0.634i)T \)
37 \( 1 + (0.940 + 0.339i)T \)
41 \( 1 + (0.998 - 0.0475i)T \)
43 \( 1 + (-0.773 + 0.634i)T \)
47 \( 1 + (-0.294 - 0.955i)T \)
53 \( 1 + (-0.557 + 0.830i)T \)
59 \( 1 + (-0.928 - 0.371i)T \)
61 \( 1 + (-0.135 - 0.990i)T \)
67 \( 1 + (-0.665 + 0.746i)T \)
71 \( 1 + (0.794 + 0.607i)T \)
73 \( 1 + (-0.396 + 0.917i)T \)
79 \( 1 + (0.628 - 0.777i)T \)
83 \( 1 + (-0.810 + 0.585i)T \)
89 \( 1 + (-0.670 + 0.742i)T \)
97 \( 1 + (0.980 - 0.195i)T \)
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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

−18.271630896756760379978760346558, −17.631609203245702576528457407938, −16.94271265522221473359079322995, −16.12708679570607038356493118719, −15.33236732541554803788133978326, −14.68925916431401017838844440366, −13.96784582183592113070638738562, −13.31612503957099723647741021166, −12.79924735947954947288895207567, −11.911090489815512615574067212130, −11.21237571210736072256879091758, −10.90950717425261316553026664394, −10.2313834914600314587754496259, −9.38590661499572767179289993876, −8.82567419368506153515963940701, −8.18674968042107241729792961137, −7.53667168136062839092318064173, −6.23168785774987570547518347604, −6.12590452241819174112212319208, −4.95481589169782957977770354664, −4.20017280993019382552498089269, −3.402171925556900554838867985379, −2.55135022343622633290337525546, −2.19759025112591413088655826625, −1.39730740969263833020555753783, 0.09406235944903323212444289771, 0.9698998375335990106239859252, 1.71097346702547723632965258155, 2.78122497271182046037682722939, 4.08253357404205701902674292945, 4.61856027789533186685592411904, 5.08023909948787641131481076785, 5.94437192278156989890705357564, 6.639580973934878663913947423629, 7.35147484739178680268080308169, 8.195801133832508711846543678555, 8.3778494837055361869824500380, 9.40744303309521835879374609804, 9.91438565670188302118385843447, 10.61465185295958056588817990849, 11.18532961145080565768031924325, 12.46317606717866180548282977062, 13.061394390511410541794893638456, 13.464295928263361458803092383489, 14.08713615168439550531740546184, 14.956786163133875111373565754767, 15.579349367664694660637016548647, 16.101573286695744517694571444172, 16.8117953923358285368394728030, 17.366415626319803574944732201529

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