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Properties

Label 1-6003-6003.1145-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.838 + 0.544i$
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  + (−0.175 − 0.984i)2-s + (−0.938 + 0.346i)4-s + (−0.833 − 0.552i)5-s + (−0.802 − 0.596i)7-s + (0.505 + 0.862i)8-s + (−0.396 + 0.917i)10-s + (−0.409 − 0.912i)11-s + (−0.464 + 0.885i)13-s + (−0.446 + 0.894i)14-s + (0.760 − 0.649i)16-s + (0.909 + 0.415i)17-s + (−0.639 + 0.768i)19-s + (0.973 + 0.229i)20-s + (−0.826 + 0.563i)22-s + (0.390 + 0.920i)25-s + (0.953 + 0.301i)26-s + ⋯
L(s)  = 1  + (−0.175 − 0.984i)2-s + (−0.938 + 0.346i)4-s + (−0.833 − 0.552i)5-s + (−0.802 − 0.596i)7-s + (0.505 + 0.862i)8-s + (−0.396 + 0.917i)10-s + (−0.409 − 0.912i)11-s + (−0.464 + 0.885i)13-s + (−0.446 + 0.894i)14-s + (0.760 − 0.649i)16-s + (0.909 + 0.415i)17-s + (−0.639 + 0.768i)19-s + (0.973 + 0.229i)20-s + (−0.826 + 0.563i)22-s + (0.390 + 0.920i)25-s + (0.953 + 0.301i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.838 + 0.544i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.838 + 0.544i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09409485653 - 0.3179648203i\)
\(L(\frac12)\) \(\approx\) \(-0.09409485653 - 0.3179648203i\)
\(L(1)\) \(\approx\) \(0.5004091893 - 0.3345246793i\)
\(L(1)\) \(\approx\) \(0.5004091893 - 0.3345246793i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.175 - 0.984i)T \)
5 \( 1 + (-0.833 - 0.552i)T \)
7 \( 1 + (-0.802 - 0.596i)T \)
11 \( 1 + (-0.409 - 0.912i)T \)
13 \( 1 + (-0.464 + 0.885i)T \)
17 \( 1 + (0.909 + 0.415i)T \)
19 \( 1 + (-0.639 + 0.768i)T \)
31 \( 1 + (0.903 + 0.427i)T \)
37 \( 1 + (0.574 - 0.818i)T \)
41 \( 1 + (-0.690 - 0.723i)T \)
43 \( 1 + (-0.903 + 0.427i)T \)
47 \( 1 + (-0.149 + 0.988i)T \)
53 \( 1 + (-0.882 - 0.470i)T \)
59 \( 1 + (-0.981 + 0.189i)T \)
61 \( 1 + (-0.0679 + 0.997i)T \)
67 \( 1 + (0.912 + 0.409i)T \)
71 \( 1 + (0.947 - 0.320i)T \)
73 \( 1 + (-0.202 - 0.979i)T \)
79 \( 1 + (0.333 + 0.942i)T \)
83 \( 1 + (0.951 + 0.307i)T \)
89 \( 1 + (-0.359 - 0.933i)T \)
97 \( 1 + (0.773 - 0.634i)T \)
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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

−18.30838577745321963414111650223, −17.271368131761661825827621612841, −16.82949550028236268632330710917, −15.91203075724623655589676676139, −15.37521069868641112720692648161, −15.163928225463812550546085191769, −14.491321510244422605692472213428, −13.54938422376553338644731955059, −12.87815620583137543041032938331, −12.330372575222743007089777299262, −11.62904481298455464578371427682, −10.55874012458130641967741206342, −9.96160946409762012107289445008, −9.53799130797334326983179311947, −8.51468238206100560957642034813, −7.94365679635849132739288826164, −7.40846399669593446918924613484, −6.61629387383961367562708919640, −6.228185546422469851497746291127, −5.03496859466334380704753312577, −4.84532534750850041717123030902, −3.67817674050988100204047191197, −3.05887385607515088720804436793, −2.252623095116761589695208392712, −0.76041124527892761586397079490, 0.1446038603393090365431991407, 1.005134124897414366123638954356, 1.82966829363989061981653409127, 2.97371027297807729934869823759, 3.47140288266190590794738077445, 4.13268379211941659755994408026, 4.7441924784506529434990708504, 5.65749556202201893059447659007, 6.52371814459337509063793460859, 7.53011037459675509725634739162, 8.07460763401745865603942304399, 8.69185048160610965287463419874, 9.452399474394280039959528717838, 10.08328354274124468791916594964, 10.73889195316668801356793835312, 11.37750450371428863579083938878, 12.13432713546770429786295341984, 12.567196149543382016821806538696, 13.168911007935768258974230259216, 13.95607181017133225700750472277, 14.44286160328096574938617080529, 15.47052174978403725446341459036, 16.32668545419934441257819981689, 16.709646093217366278327279474578, 17.132998604260249823040115786837

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