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Properties

Label 1-6027-6027.4514-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.999 + 0.0293i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (0.995 + 0.0896i)2-s + (0.983 + 0.178i)4-s + (−0.0448 − 0.998i)5-s + (0.963 + 0.266i)8-s + (0.0448 − 0.998i)10-s + (0.753 + 0.657i)11-s + (−0.995 − 0.0896i)13-s + (0.936 + 0.351i)16-s + (−0.983 + 0.178i)17-s + (0.309 + 0.951i)19-s + (0.134 − 0.990i)20-s + (0.691 + 0.722i)22-s + (−0.134 − 0.990i)23-s + (−0.995 + 0.0896i)25-s + (−0.983 − 0.178i)26-s + ⋯
L(s)  = 1  + (0.995 + 0.0896i)2-s + (0.983 + 0.178i)4-s + (−0.0448 − 0.998i)5-s + (0.963 + 0.266i)8-s + (0.0448 − 0.998i)10-s + (0.753 + 0.657i)11-s + (−0.995 − 0.0896i)13-s + (0.936 + 0.351i)16-s + (−0.983 + 0.178i)17-s + (0.309 + 0.951i)19-s + (0.134 − 0.990i)20-s + (0.691 + 0.722i)22-s + (−0.134 − 0.990i)23-s + (−0.995 + 0.0896i)25-s + (−0.983 − 0.178i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.999 + 0.0293i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (4514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.999 + 0.0293i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.808813865 + 0.05588568130i\)
\(L(\frac12)\) \(\approx\) \(3.808813865 + 0.05588568130i\)
\(L(1)\) \(\approx\) \(2.029597408 - 0.04451206444i\)
\(L(1)\) \(\approx\) \(2.029597408 - 0.04451206444i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.995 + 0.0896i)T \)
5 \( 1 + (-0.0448 - 0.998i)T \)
11 \( 1 + (0.753 + 0.657i)T \)
13 \( 1 + (-0.995 - 0.0896i)T \)
17 \( 1 + (-0.983 + 0.178i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.134 - 0.990i)T \)
29 \( 1 + (0.473 + 0.880i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.473 + 0.880i)T \)
43 \( 1 + (-0.550 - 0.834i)T \)
47 \( 1 + (0.995 + 0.0896i)T \)
53 \( 1 + (0.983 + 0.178i)T \)
59 \( 1 + (-0.550 - 0.834i)T \)
61 \( 1 + (-0.134 + 0.990i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 + (0.473 - 0.880i)T \)
73 \( 1 + (0.222 - 0.974i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.995 - 0.0896i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−17.46126149490166617986244702738, −17.20028683320197382881571431887, −16.06700007612520588935790667615, −15.60442049540848770004568378242, −15.00660274007316853879908514686, −14.35116431331078074430844127903, −13.78185456874024746917004311577, −13.37310714702539142261751833419, −12.397219369910886286366587615265, −11.56995154686969489839373226806, −11.43642921196917966563090479137, −10.6314575667914950755918500640, −9.87308944304493770814860607181, −9.21510084959003692996034277225, −8.134691815952966958640603002414, −7.29450160862277540673046156319, −6.849529218672715570260628239343, −6.22524067373469739452489133989, −5.51347183293613976965530012303, −4.62134212761394726259740044055, −4.02989584553852805635050161669, −3.19115478736615141587631576802, −2.60895071930472804629506056842, −1.97872324814446984751688385639, −0.77194787211039964707333566142, 0.87835025907968512748749830150, 1.82877660218040276884614020289, 2.389156044570607775609205617157, 3.42050798374054056873654513495, 4.36846207136637596408675140855, 4.5404982292932132943559228499, 5.348461584770562809909638476896, 6.14410949258385591593714273184, 6.805733915422385273713843141106, 7.51451181643573092198107374407, 8.290121438980348323485183404648, 8.95074519120961459127843247982, 9.89821833989345764913508479584, 10.382160898001212730447045114249, 11.45914861925097442170554880238, 12.0504486255010922326048754050, 12.43019316996385477402928921705, 13.01311766215867487323030072821, 13.83615989150502097319960820941, 14.33803680268097753327996323077, 15.177566672168090289605998487698, 15.50227529488443759496436918355, 16.52500372967430396367014351558, 16.82616891727003075689970245159, 17.398916314028458410322517285322

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