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Properties

Label 1-6039-6039.3625-r0-0-0
Degree $1$
Conductor $6039$
Sign $-0.228 - 0.973i$
Analytic cond. $28.0449$
Root an. cond. $28.0449$
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  + i·2-s − 4-s + (0.978 − 0.207i)5-s + (−0.951 + 0.309i)7-s − i·8-s + (0.207 + 0.978i)10-s + (−0.913 − 0.406i)13-s + (−0.309 − 0.951i)14-s + 16-s + (0.207 + 0.978i)17-s + (−0.978 + 0.207i)19-s + (−0.978 + 0.207i)20-s + (−0.207 + 0.978i)23-s + (0.913 − 0.406i)25-s + (0.406 − 0.913i)26-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (0.978 − 0.207i)5-s + (−0.951 + 0.309i)7-s − i·8-s + (0.207 + 0.978i)10-s + (−0.913 − 0.406i)13-s + (−0.309 − 0.951i)14-s + 16-s + (0.207 + 0.978i)17-s + (−0.978 + 0.207i)19-s + (−0.978 + 0.207i)20-s + (−0.207 + 0.978i)23-s + (0.913 − 0.406i)25-s + (0.406 − 0.913i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
Sign: $-0.228 - 0.973i$
Analytic conductor: \(28.0449\)
Root analytic conductor: \(28.0449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6039} (3625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6039,\ (0:\ ),\ -0.228 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04948432655 + 0.06246142898i\)
\(L(\frac12)\) \(\approx\) \(-0.04948432655 + 0.06246142898i\)
\(L(1)\) \(\approx\) \(0.7181296117 + 0.4052446040i\)
\(L(1)\) \(\approx\) \(0.7181296117 + 0.4052446040i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
61 \( 1 \)
good2 \( 1 + iT \)
5 \( 1 + (0.978 - 0.207i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
13 \( 1 + (-0.913 - 0.406i)T \)
17 \( 1 + (0.207 + 0.978i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.207 + 0.978i)T \)
29 \( 1 + (0.743 - 0.669i)T \)
31 \( 1 + (0.951 + 0.309i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.951 - 0.309i)T \)
47 \( 1 + (-0.978 - 0.207i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (-0.406 + 0.913i)T \)
67 \( 1 + (0.743 + 0.669i)T \)
71 \( 1 + (-0.406 + 0.913i)T \)
73 \( 1 + (0.104 - 0.994i)T \)
79 \( 1 + (-0.951 + 0.309i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (0.587 - 0.809i)T \)
97 \( 1 + (-0.669 - 0.743i)T \)
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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

−17.398692390099988396986840689316, −16.71017384098521614522186505542, −16.18956604064139651083473304418, −15.00811701417191764473397402965, −14.38479435027005810802182437642, −13.796747708097349353113339059919, −13.30375909430916888351592359793, −12.46744116293351452006559413967, −12.22191152033652847178394412390, −11.14231164288295918024034117890, −10.54387856971210453785903925960, −9.87799899063507454626791330265, −9.57505929963294962860744410417, −8.84727395624179895516592316644, −8.05340747194841909649465546619, −6.86716486309659982248292391457, −6.5517043373459246982190294849, −5.54770989655365745788013278355, −4.8026284890381992973572944811, −4.21820885532830721240961367863, −3.11197898985155612721745778687, −2.68091708932905775133072292351, −2.028325394208347096919143370282, −1.02369665892132243929923805096, −0.02190311827841381807428640778, 1.21024558523887878936072403595, 2.21456032024655641849037395991, 3.06883991655341541981083040643, 3.92941731121129495713486190396, 4.77429733275969052129784325020, 5.52211318681759899687542695708, 6.034876537073225164149555326936, 6.59822173019057986982507845739, 7.27677184450903534817520441193, 8.280888606827751720390429051065, 8.69529990207057960566819372269, 9.52773160953779369523672539369, 10.185048650559187609336521576956, 10.30868505574899487261055719228, 11.9004589505645456036030848032, 12.49819791428302924656753461162, 13.10974689862417117259035439759, 13.54706899110928384050374766045, 14.35115971714904649291447939645, 14.96907673941171129890464514238, 15.5585467495346452282872025729, 16.21933634427956474211210205683, 17.04008367511627146686000316823, 17.28522446391373893327203788098, 17.87834098726498835619475698387

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