Properties

Label 1-6027-6027.5312-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.568 - 0.822i$
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  + (0.963 + 0.266i)2-s + (0.858 + 0.512i)4-s + (0.134 + 0.990i)5-s + (0.691 + 0.722i)8-s + (−0.134 + 0.990i)10-s + (−0.550 + 0.834i)11-s + (−0.963 − 0.266i)13-s + (0.473 + 0.880i)16-s + (−0.858 + 0.512i)17-s + (−0.809 − 0.587i)19-s + (−0.393 + 0.919i)20-s + (−0.753 + 0.657i)22-s + (0.393 + 0.919i)23-s + (−0.963 + 0.266i)25-s + (−0.858 − 0.512i)26-s + ⋯
L(s)  = 1  + (0.963 + 0.266i)2-s + (0.858 + 0.512i)4-s + (0.134 + 0.990i)5-s + (0.691 + 0.722i)8-s + (−0.134 + 0.990i)10-s + (−0.550 + 0.834i)11-s + (−0.963 − 0.266i)13-s + (0.473 + 0.880i)16-s + (−0.858 + 0.512i)17-s + (−0.809 − 0.587i)19-s + (−0.393 + 0.919i)20-s + (−0.753 + 0.657i)22-s + (0.393 + 0.919i)23-s + (−0.963 + 0.266i)25-s + (−0.858 − 0.512i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.568 - 0.822i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (5312, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.568 - 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4051909646 + 0.7726229050i\)
\(L(\frac12)\) \(\approx\) \(-0.4051909646 + 0.7726229050i\)
\(L(1)\) \(\approx\) \(1.253562758 + 0.7175803543i\)
\(L(1)\) \(\approx\) \(1.253562758 + 0.7175803543i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.963 + 0.266i)T \)
5 \( 1 + (0.134 + 0.990i)T \)
11 \( 1 + (-0.550 + 0.834i)T \)
13 \( 1 + (-0.963 - 0.266i)T \)
17 \( 1 + (-0.858 + 0.512i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.393 + 0.919i)T \)
29 \( 1 + (-0.995 - 0.0896i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.995 - 0.0896i)T \)
43 \( 1 + (0.983 - 0.178i)T \)
47 \( 1 + (0.963 + 0.266i)T \)
53 \( 1 + (0.858 + 0.512i)T \)
59 \( 1 + (0.983 - 0.178i)T \)
61 \( 1 + (0.393 - 0.919i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (-0.995 + 0.0896i)T \)
73 \( 1 + (-0.623 + 0.781i)T \)
79 \( 1 - T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (0.963 - 0.266i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−17.06011698191545329250223273552, −16.37872692041494667936517020236, −16.10459846881948407637338332786, −15.19134693497047284981205886371, −14.59823473008136551843141283331, −13.87437438171676337492997620107, −13.26493280259012329829912441361, −12.75051364652544434142928312621, −12.14240838732513859300509515044, −11.56061316085273124742108768022, −10.63066357993493413443264761517, −10.2987610881692640197700368581, −9.165764582645139218215410324981, −8.75618892541326065865476388431, −7.784391383225898317645643269560, −7.04714390786224544440680508146, −6.29420690139334125814549368698, −5.45665587809535143600088766397, −5.06428116968164098628157718707, −4.30676591769824340975797645112, −3.68768462564736803066699024142, −2.59387145984324232751547970159, −2.15182958281608354310124402134, −1.16275782672267246580435734619, −0.131924704839663152166955684474, 1.82307125488471798330101604117, 2.33850469877662651222244335761, 2.89141153891949923315985311608, 3.92143038676657522689449380228, 4.37784317280176986927917103971, 5.37348510341781519472794317412, 5.81469430764955048771694658068, 6.78589026367275756914423418852, 7.259046730743207254461032412085, 7.657280863074035851823977846674, 8.72199011807129043796129324706, 9.639461436579466205970261196292, 10.43302368666358636348998025796, 10.91726496486391993458313021592, 11.59070574184105856676952745201, 12.333757494940192396287155565941, 13.19303438492065290071986316091, 13.29802410707064485138748050243, 14.48721696168028778430564634657, 14.748306239572585768804384044969, 15.49937644830541690123721048930, 15.67004841313408734394004496486, 16.98476348718223735755543227358, 17.39919007311109233546571648122, 17.84289212927904310894819052297

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