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Properties

Label 1-6027-6027.809-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.584 + 0.811i$
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  + (−0.611 + 0.791i)2-s + (−0.251 − 0.967i)4-s + (0.323 + 0.946i)5-s + (0.919 + 0.393i)8-s + (−0.946 − 0.323i)10-s + (−0.973 + 0.229i)11-s + (−0.795 + 0.605i)13-s + (−0.873 + 0.486i)16-s + (0.999 − 0.00747i)17-s + (−0.998 − 0.0523i)19-s + (0.834 − 0.550i)20-s + (0.413 − 0.910i)22-s + (−0.447 − 0.894i)23-s + (−0.791 + 0.611i)25-s + (0.00747 − 0.999i)26-s + ⋯
L(s)  = 1  + (−0.611 + 0.791i)2-s + (−0.251 − 0.967i)4-s + (0.323 + 0.946i)5-s + (0.919 + 0.393i)8-s + (−0.946 − 0.323i)10-s + (−0.973 + 0.229i)11-s + (−0.795 + 0.605i)13-s + (−0.873 + 0.486i)16-s + (0.999 − 0.00747i)17-s + (−0.998 − 0.0523i)19-s + (0.834 − 0.550i)20-s + (0.413 − 0.910i)22-s + (−0.447 − 0.894i)23-s + (−0.791 + 0.611i)25-s + (0.00747 − 0.999i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.584 + 0.811i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.584 + 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7139733580 + 0.3657201073i\)
\(L(\frac12)\) \(\approx\) \(0.7139733580 + 0.3657201073i\)
\(L(1)\) \(\approx\) \(0.6078514786 + 0.3036376471i\)
\(L(1)\) \(\approx\) \(0.6078514786 + 0.3036376471i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.611 + 0.791i)T \)
5 \( 1 + (0.323 + 0.946i)T \)
11 \( 1 + (-0.973 + 0.229i)T \)
13 \( 1 + (-0.795 + 0.605i)T \)
17 \( 1 + (0.999 - 0.00747i)T \)
19 \( 1 + (-0.998 - 0.0523i)T \)
23 \( 1 + (-0.447 - 0.894i)T \)
29 \( 1 + (-0.738 + 0.674i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (-0.842 - 0.538i)T \)
43 \( 1 + (0.0896 + 0.995i)T \)
47 \( 1 + (0.991 - 0.126i)T \)
53 \( 1 + (0.506 - 0.862i)T \)
59 \( 1 + (-0.575 - 0.817i)T \)
61 \( 1 + (-0.0598 - 0.998i)T \)
67 \( 1 + (-0.933 + 0.358i)T \)
71 \( 1 + (0.674 - 0.738i)T \)
73 \( 1 + (0.563 - 0.826i)T \)
79 \( 1 + (-0.965 + 0.258i)T \)
83 \( 1 + (0.900 - 0.433i)T \)
89 \( 1 + (-0.386 - 0.922i)T \)
97 \( 1 + (-0.156 + 0.987i)T \)
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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

−17.63583721771802512920744884740, −16.94557464005566220347458312583, −16.695048668568969082258822829031, −15.71580535423279397312071537790, −15.23592060917028466567137032824, −13.989574508311344678651712956007, −13.55037514489969274631549537234, −12.767218612072980899249418519899, −12.34479029314867059212762291091, −11.821956903036469280058997627788, −10.80791850452930828968349047213, −10.199482060968172455629712147241, −9.83034207385861937484106478242, −8.91386278003286740457028777250, −8.42795934890809385834008437753, −7.7058644568596866203638026111, −7.22629608778599178813896760833, −5.84601878582134938982920604053, −5.357672560620661240426676147515, −4.56322617299308310240581633784, −3.781769370412319754436694475741, −2.919803497836529669937092730754, −2.21187164776677544076198177683, −1.45257484080799624527061784105, −0.55433133842471245548957131547, 0.41610477377283568821860850332, 1.930531042987003547667557952985, 2.20716141937337318702337347321, 3.281282024190899702263408347134, 4.31876236720359251783418676401, 5.06615795039927381287372272980, 5.80658823126500541039862074057, 6.415216214562292066478620404, 7.14240807230559814918809238160, 7.65159163631620576491204828002, 8.27883110291388230562000721109, 9.253851722159257577553464295843, 9.77061536107566626083894105735, 10.51023105792637959038393538, 10.778439657678892971662739753415, 11.76781451700049606526829374838, 12.66857622587329740584377613836, 13.36892954598318687969301473966, 14.30995039062625497822043876522, 14.50563345563317996661669053896, 15.18709288343348188551829434936, 15.82658868465678275332049422020, 16.683032485959980700186881559281, 17.04492018453657876539350381979, 17.85708033858484228105561405372

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