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Properties

Label 1-6027-6027.1004-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.585 - 0.810i$
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  + (−0.999 − 0.0299i)2-s + (0.998 + 0.0598i)4-s + (−0.0149 − 0.999i)5-s + (−0.995 − 0.0896i)8-s + (−0.0149 + 0.999i)10-s + (−0.237 + 0.971i)11-s + (−0.880 + 0.473i)13-s + (0.992 + 0.119i)16-s + (−0.894 − 0.447i)17-s + (−0.994 + 0.104i)19-s + (0.0448 − 0.998i)20-s + (0.266 − 0.963i)22-s + (−0.887 − 0.460i)23-s + (−0.999 + 0.0299i)25-s + (0.894 − 0.447i)26-s + ⋯
L(s)  = 1  + (−0.999 − 0.0299i)2-s + (0.998 + 0.0598i)4-s + (−0.0149 − 0.999i)5-s + (−0.995 − 0.0896i)8-s + (−0.0149 + 0.999i)10-s + (−0.237 + 0.971i)11-s + (−0.880 + 0.473i)13-s + (0.992 + 0.119i)16-s + (−0.894 − 0.447i)17-s + (−0.994 + 0.104i)19-s + (0.0448 − 0.998i)20-s + (0.266 − 0.963i)22-s + (−0.887 − 0.460i)23-s + (−0.999 + 0.0299i)25-s + (0.894 − 0.447i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.585 - 0.810i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (1004, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.585 - 0.810i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3991171457 - 0.2041154580i\)
\(L(\frac12)\) \(\approx\) \(0.3991171457 - 0.2041154580i\)
\(L(1)\) \(\approx\) \(0.5229714629 - 0.05241281020i\)
\(L(1)\) \(\approx\) \(0.5229714629 - 0.05241281020i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.999 - 0.0299i)T \)
5 \( 1 + (-0.0149 - 0.999i)T \)
11 \( 1 + (-0.237 + 0.971i)T \)
13 \( 1 + (-0.880 + 0.473i)T \)
17 \( 1 + (-0.894 - 0.447i)T \)
19 \( 1 + (-0.994 + 0.104i)T \)
23 \( 1 + (-0.887 - 0.460i)T \)
29 \( 1 + (-0.351 + 0.936i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (-0.772 + 0.635i)T \)
43 \( 1 + (-0.753 + 0.657i)T \)
47 \( 1 + (0.0299 - 0.999i)T \)
53 \( 1 + (-0.0598 + 0.998i)T \)
59 \( 1 + (-0.946 - 0.323i)T \)
61 \( 1 + (-0.842 - 0.538i)T \)
67 \( 1 + (0.743 + 0.669i)T \)
71 \( 1 + (0.351 + 0.936i)T \)
73 \( 1 + (0.826 + 0.563i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (0.850 - 0.525i)T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−17.66939437833457655441284762272, −17.40493532843650632842201912001, −16.66073332254232378245547816323, −15.73581097955537009468576580397, −15.36979112391855333422233061149, −14.770084659481469288050791905721, −13.95856527800358460480203904183, −13.28730684121791045201220109435, −12.27565205908366270552921179223, −11.718925647000316974478019205681, −10.85353385300384067500918473328, −10.630041506863482321428202292234, −9.90209661573140986141123732932, −9.19356771245054580410336252445, −8.32777789417522469915157286339, −7.87259436479092757363451892518, −7.17463825745282017099506414995, −6.241512971138556324102669630385, −6.135628463955210476145339630427, −4.99228077341357822755517918632, −3.84132237120334624998513204807, −3.15864767738181974235310757645, −2.332312367286235164454343795085, −1.906930226975225534321874373098, −0.469103166966434588242458288309, 0.28979010511722876764416215395, 1.528934917532822553597451085027, 2.047959621876980400793087398209, 2.747645043220076439644678209235, 4.01238417347322230930674339016, 4.671421899654953574853721080332, 5.3236411322638872662852880581, 6.43495052868645943041311584069, 6.87034200751170611829858986173, 7.72917316125192815648610861597, 8.342139850315135520451224662153, 8.93603711125278889127801520997, 9.63958710889996596352821409030, 10.07383440596847211152511404849, 10.88141442274274429091555996282, 11.73002628355132750430545118192, 12.28701500344172486614860520765, 12.70687908547386836610237554389, 13.61242764140256655582823388591, 14.51308864138602783106081495405, 15.286768194795340583100413081552, 15.72323362411833127982387043875, 16.48458433697562813591322141942, 17.14252955591876868505807573100, 17.37506853116993258273634337862

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