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Properties

Label 1-6039-6039.3056-r0-0-0
Degree $1$
Conductor $6039$
Sign $0.169 + 0.985i$
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  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (0.669 + 0.743i)5-s + i·7-s + (−0.587 − 0.809i)8-s + (−0.406 − 0.913i)10-s + (0.913 − 0.406i)13-s + (0.309 − 0.951i)14-s + (0.309 + 0.951i)16-s + (−0.743 + 0.669i)17-s + (0.5 − 0.866i)19-s + (0.104 + 0.994i)20-s + (0.406 − 0.913i)23-s + (−0.104 + 0.994i)25-s + (−0.994 + 0.104i)26-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (0.669 + 0.743i)5-s + i·7-s + (−0.587 − 0.809i)8-s + (−0.406 − 0.913i)10-s + (0.913 − 0.406i)13-s + (0.309 − 0.951i)14-s + (0.309 + 0.951i)16-s + (−0.743 + 0.669i)17-s + (0.5 − 0.866i)19-s + (0.104 + 0.994i)20-s + (0.406 − 0.913i)23-s + (−0.104 + 0.994i)25-s + (−0.994 + 0.104i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
Sign: $0.169 + 0.985i$
Analytic conductor: \(28.0449\)
Root analytic conductor: \(28.0449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6039} (3056, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6039,\ (0:\ ),\ 0.169 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.069951348 + 0.9013298371i\)
\(L(\frac12)\) \(\approx\) \(1.069951348 + 0.9013298371i\)
\(L(1)\) \(\approx\) \(0.8508854116 + 0.1818539961i\)
\(L(1)\) \(\approx\) \(0.8508854116 + 0.1818539961i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T \)
5 \( 1 + (0.669 + 0.743i)T \)
7 \( 1 + iT \)
13 \( 1 + (0.913 - 0.406i)T \)
17 \( 1 + (-0.743 + 0.669i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.406 - 0.913i)T \)
29 \( 1 + (0.743 + 0.669i)T \)
31 \( 1 + (0.587 + 0.809i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (-0.587 + 0.809i)T \)
47 \( 1 + (0.978 - 0.207i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (0.994 - 0.104i)T \)
67 \( 1 + (-0.994 + 0.104i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + (0.669 + 0.743i)T \)
79 \( 1 + (-0.951 + 0.309i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (-0.951 + 0.309i)T \)
97 \( 1 + (-0.913 - 0.406i)T \)
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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

−17.64074976181835930217765508153, −16.77107615151149507648401338672, −16.58635874664929093752529259985, −15.7373839222382057678371376919, −15.26480375281104908382010866891, −14.06542897225429606009022552525, −13.73234834688590833805990125353, −13.22727431193121347995237609851, −12.01754074570857433821764312417, −11.581408132894214135466860492335, −10.72727621759452250903845786343, −10.07562203077714780760681779310, −9.62637148829245292300008385671, −8.818031317338592885361812096722, −8.368035830662417887457019214867, −7.49191511110518859033961786969, −6.91546898676452857567627463970, −6.1267277802803325944621874526, −5.56889844591672268653858833453, −4.66678694843102242296162462629, −3.90977322618541374086431730476, −2.85228179221345503458323225248, −1.88428560119084756401725047868, −1.24622771862908998207209294499, −0.56339710573562562410892287649, 0.97827064676887970228674467973, 1.76770633481937087980076601874, 2.70841904854480907136459280407, 2.87140441664896480551821478660, 3.91707532930442408790576468307, 5.08098674705255068299817074499, 5.90582136212794304253472901436, 6.55599593250079104800673603147, 6.959855839735920986193664155626, 8.04237020244914421518542024268, 8.74351788097285543997652933868, 9.02734506920008149760363693975, 9.96621454065422038174733076561, 10.534153149419573965651614693820, 11.1066187440054040088351903101, 11.66548917756916364960722546461, 12.58707994413941806142305009940, 13.07288420012317293436870125337, 13.88134525955824759735078291426, 14.85871651687913175959605204679, 15.26987467690535429369696573484, 15.978260332624582033082191918334, 16.563324247676621942487663701762, 17.64366440635871778976343972678, 17.88374588971913644482298642082

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