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Properties

Label 1-6027-6027.95-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.0215 - 0.999i$
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  + (−0.119 + 0.992i)2-s + (−0.971 − 0.237i)4-s + (−0.0598 − 0.998i)5-s + (0.351 − 0.936i)8-s + (0.998 + 0.0598i)10-s + (0.171 − 0.985i)11-s + (−0.372 + 0.928i)13-s + (0.887 + 0.460i)16-s + (0.480 + 0.877i)17-s + (−0.358 + 0.933i)19-s + (−0.178 + 0.983i)20-s + (0.957 + 0.287i)22-s + (−0.337 + 0.941i)23-s + (−0.992 + 0.119i)25-s + (−0.877 − 0.480i)26-s + ⋯
L(s)  = 1  + (−0.119 + 0.992i)2-s + (−0.971 − 0.237i)4-s + (−0.0598 − 0.998i)5-s + (0.351 − 0.936i)8-s + (0.998 + 0.0598i)10-s + (0.171 − 0.985i)11-s + (−0.372 + 0.928i)13-s + (0.887 + 0.460i)16-s + (0.480 + 0.877i)17-s + (−0.358 + 0.933i)19-s + (−0.178 + 0.983i)20-s + (0.957 + 0.287i)22-s + (−0.337 + 0.941i)23-s + (−0.992 + 0.119i)25-s + (−0.877 − 0.480i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0215 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0215 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.0215 - 0.999i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.0215 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2365311736 - 0.2314855082i\)
\(L(\frac12)\) \(\approx\) \(0.2365311736 - 0.2314855082i\)
\(L(1)\) \(\approx\) \(0.7129124644 + 0.2348151582i\)
\(L(1)\) \(\approx\) \(0.7129124644 + 0.2348151582i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.119 + 0.992i)T \)
5 \( 1 + (-0.0598 - 0.998i)T \)
11 \( 1 + (0.171 - 0.985i)T \)
13 \( 1 + (-0.372 + 0.928i)T \)
17 \( 1 + (0.480 + 0.877i)T \)
19 \( 1 + (-0.358 + 0.933i)T \)
23 \( 1 + (-0.337 + 0.941i)T \)
29 \( 1 + (-0.605 + 0.795i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.925 - 0.379i)T \)
43 \( 1 + (-0.266 + 0.963i)T \)
47 \( 1 + (0.786 - 0.617i)T \)
53 \( 1 + (-0.519 - 0.854i)T \)
59 \( 1 + (-0.251 - 0.967i)T \)
61 \( 1 + (0.762 + 0.646i)T \)
67 \( 1 + (0.544 - 0.838i)T \)
71 \( 1 + (0.795 - 0.605i)T \)
73 \( 1 + (0.680 + 0.733i)T \)
79 \( 1 + (-0.965 + 0.258i)T \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + (0.141 - 0.989i)T \)
97 \( 1 + (-0.453 - 0.891i)T \)
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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

−18.01836454939670221942824943077, −17.37215985845207511729401151525, −16.946580063033818658510630284925, −15.694574099705665892032312017828, −15.15426345612242390386712662125, −14.495930775095043876003231542132, −13.91737096179009089670106744387, −13.14826606870945231827574896140, −12.50018155827144463243560018045, −11.87289758774715421155991501001, −11.256112041518070259779701351929, −10.52140425314784961217694828971, −10.078103783774982535203293273465, −9.46041658842410252336864929500, −8.7148054351593500785059219821, −7.66419025153670028319280443807, −7.398102402652507714693932825039, −6.43664005676126436952986455001, −5.48854062588523355725346405155, −4.783655103061826635524095930219, −4.002370700963211700345815018458, −3.302359033227074517027601223380, −2.36529503825237914761202078863, −2.26245580269330385473439131573, −0.88061757585361805042847759345, 0.10480098101215193201484278007, 1.34339078134686983991815723479, 1.80593111837908695946965767967, 3.64013663051708757388483356929, 3.72089696825304358896222731409, 4.79981707737393148152221828216, 5.449549061048055846710350543351, 5.940216705544632248111679770090, 6.75216564938988476067516591718, 7.53583837899348135472479435967, 8.28854176352729691693905874531, 8.661488883849075730612012530298, 9.390112516862947124036271293422, 9.95499058344939030653244548595, 10.87668495980821197509785029457, 11.71559005777763733174157337106, 12.597612467111446689251995983473, 12.86566952782841888981002051396, 13.98786450876603315652488272560, 14.123188523094318610985308510753, 14.9797525581104059401186992427, 15.79249018216750270809554512686, 16.34637361957848281675608495460, 16.81151390707889812516708962120, 17.20725580133623030506684884380

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