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Properties

Label 1-6003-6003.1570-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.102 + 0.994i$
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  + (0.966 + 0.255i)2-s + (0.869 + 0.494i)4-s + (0.938 + 0.346i)5-s + (−0.0882 − 0.996i)7-s + (0.714 + 0.699i)8-s + (0.818 + 0.574i)10-s + (−0.994 + 0.108i)11-s + (−0.833 + 0.552i)13-s + (0.169 − 0.985i)14-s + (0.511 + 0.859i)16-s + (0.142 + 0.989i)17-s + (0.862 − 0.505i)19-s + (0.644 + 0.764i)20-s + (−0.988 − 0.149i)22-s + (0.760 + 0.649i)25-s + (−0.947 + 0.320i)26-s + ⋯
L(s)  = 1  + (0.966 + 0.255i)2-s + (0.869 + 0.494i)4-s + (0.938 + 0.346i)5-s + (−0.0882 − 0.996i)7-s + (0.714 + 0.699i)8-s + (0.818 + 0.574i)10-s + (−0.994 + 0.108i)11-s + (−0.833 + 0.552i)13-s + (0.169 − 0.985i)14-s + (0.511 + 0.859i)16-s + (0.142 + 0.989i)17-s + (0.862 − 0.505i)19-s + (0.644 + 0.764i)20-s + (−0.988 − 0.149i)22-s + (0.760 + 0.649i)25-s + (−0.947 + 0.320i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.102 + 0.994i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1570, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.102 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.845496920 + 2.566165817i\)
\(L(\frac12)\) \(\approx\) \(2.845496920 + 2.566165817i\)
\(L(1)\) \(\approx\) \(1.998171021 + 0.6332316314i\)
\(L(1)\) \(\approx\) \(1.998171021 + 0.6332316314i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.966 + 0.255i)T \)
5 \( 1 + (0.938 + 0.346i)T \)
7 \( 1 + (-0.0882 - 0.996i)T \)
11 \( 1 + (-0.994 + 0.108i)T \)
13 \( 1 + (-0.833 + 0.552i)T \)
17 \( 1 + (0.142 + 0.989i)T \)
19 \( 1 + (0.862 - 0.505i)T \)
31 \( 1 + (-0.601 - 0.798i)T \)
37 \( 1 + (-0.794 + 0.607i)T \)
41 \( 1 + (0.995 - 0.0950i)T \)
43 \( 1 + (-0.601 + 0.798i)T \)
47 \( 1 + (0.733 + 0.680i)T \)
53 \( 1 + (0.742 + 0.670i)T \)
59 \( 1 + (0.723 + 0.690i)T \)
61 \( 1 + (-0.810 - 0.585i)T \)
67 \( 1 + (0.994 + 0.108i)T \)
71 \( 1 + (0.882 - 0.470i)T \)
73 \( 1 + (0.301 + 0.953i)T \)
79 \( 1 + (0.999 + 0.0135i)T \)
83 \( 1 + (-0.546 - 0.837i)T \)
89 \( 1 + (0.523 + 0.852i)T \)
97 \( 1 + (0.275 - 0.961i)T \)
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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

−17.68603746204062758712008082400, −16.68663773181830754536088106037, −16.06440761836505527682663045855, −15.60743761994840465012171461049, −14.79819460543812459314103576400, −14.20448510231425704236151003799, −13.59423544906642679806847109856, −12.9549866295182414783310580937, −12.29685962169822482147938502676, −12.00201193875096573571413184256, −10.974702418854949227254092764310, −10.29611306366122965409605188389, −9.71739936491675219306118154926, −9.091891125264794852310767654055, −8.110612433294223264691868527667, −7.317018272980460725336965663075, −6.6076546971791732697310048773, −5.58893503641674128700752139553, −5.29150109937715005991017655670, −5.03925916457617438094938976158, −3.7291011872108310962118387904, −2.87024300234947843376373874129, −2.430704163180491233556366658563, −1.75882172786980886882706691069, −0.61533023616693817738720463487, 1.15467816730478529675588863728, 2.07526905836405427674753045132, 2.67439478991226789427400649668, 3.45921463705720145042296239067, 4.273133301790448398767803285096, 4.989735963486600525865186591105, 5.595939151571625644445918294699, 6.330271014553718614720797885499, 7.06556229698278472428909703434, 7.49161497711569041829055894244, 8.2500108698584514318309079228, 9.44146673086134507557020554202, 9.99362097131794689032755834769, 10.7686100978816920884708965701, 11.11520943643876825785717227696, 12.196903344086031011092478103520, 12.7774127724770869359172222771, 13.51362095522720619244271091028, 13.75347912333029639896360696330, 14.568807326458264303437617535714, 15.01538356624737846017661422211, 15.87611288761723049757021003692, 16.550486458200282852122405349161, 17.17042504816478107753609805701, 17.55967547491998232312738733699

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