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Properties

Label 1-6017-6017.329-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.919 + 0.393i$
Analytic cond. $27.9428$
Root an. cond. $27.9428$
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  + (0.449 + 0.893i)2-s + (0.900 − 0.433i)3-s + (−0.596 + 0.802i)4-s + (0.985 − 0.171i)5-s + (0.792 + 0.609i)6-s + (0.770 + 0.636i)7-s + (−0.985 − 0.171i)8-s + (0.623 − 0.781i)9-s + (0.596 + 0.802i)10-s + (−0.188 + 0.982i)12-s + (0.222 − 0.974i)13-s + (−0.222 + 0.974i)14-s + (0.813 − 0.582i)15-s + (−0.289 − 0.957i)16-s + (−0.952 − 0.305i)17-s + (0.978 + 0.205i)18-s + ⋯
L(s)  = 1  + (0.449 + 0.893i)2-s + (0.900 − 0.433i)3-s + (−0.596 + 0.802i)4-s + (0.985 − 0.171i)5-s + (0.792 + 0.609i)6-s + (0.770 + 0.636i)7-s + (−0.985 − 0.171i)8-s + (0.623 − 0.781i)9-s + (0.596 + 0.802i)10-s + (−0.188 + 0.982i)12-s + (0.222 − 0.974i)13-s + (−0.222 + 0.974i)14-s + (0.813 − 0.582i)15-s + (−0.289 − 0.957i)16-s + (−0.952 − 0.305i)17-s + (0.978 + 0.205i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.919 + 0.393i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.919 + 0.393i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.392577762 + 0.9002216442i\)
\(L(\frac12)\) \(\approx\) \(4.392577762 + 0.9002216442i\)
\(L(1)\) \(\approx\) \(2.143984297 + 0.5962184960i\)
\(L(1)\) \(\approx\) \(2.143984297 + 0.5962184960i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.449 + 0.893i)T \)
3 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (0.985 - 0.171i)T \)
7 \( 1 + (0.770 + 0.636i)T \)
13 \( 1 + (0.222 - 0.974i)T \)
17 \( 1 + (-0.952 - 0.305i)T \)
19 \( 1 + (0.928 - 0.370i)T \)
23 \( 1 + (0.994 + 0.103i)T \)
29 \( 1 + (-0.851 + 0.524i)T \)
31 \( 1 + (0.154 - 0.987i)T \)
37 \( 1 + (0.700 + 0.713i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.999 + 0.0345i)T \)
47 \( 1 + (0.120 - 0.992i)T \)
53 \( 1 + (0.813 - 0.582i)T \)
59 \( 1 + (-0.568 + 0.822i)T \)
61 \( 1 + (-0.418 + 0.908i)T \)
67 \( 1 + (0.188 - 0.982i)T \)
71 \( 1 + (-0.997 - 0.0689i)T \)
73 \( 1 + (-0.813 - 0.582i)T \)
79 \( 1 + (-0.999 + 0.0345i)T \)
83 \( 1 + (-0.748 - 0.663i)T \)
89 \( 1 + (0.418 + 0.908i)T \)
97 \( 1 + (0.990 - 0.137i)T \)
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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

−17.77686664014607183193615884689, −17.16825065534277376772848182565, −16.3132192436262399332147866664, −15.46559868015368805219846447810, −14.632316500145880857356539639130, −14.29879966472859860741138945482, −13.79246513023176487294889122906, −13.21193181932605758840622811715, −12.6795044379030525288522694379, −11.40585656211017933089512869331, −11.11669700624435447055361195081, −10.34701885644778434170999169302, −9.81520515514933519412138909230, −9.026573821310962956141372553044, −8.78607065719161450804886181736, −7.64200946709817555227531923486, −6.89523579808905609089134160542, −5.98497135356617031891639308830, −5.13477128530683322311856602742, −4.51279951972965146492752395695, −3.96478687185336363299195221683, −3.097409995993462351107856597108, −2.39640815115011059792990797894, −1.67368220831097225891894539829, −1.20994241897265398279474382992, 0.845357792852876081646973552423, 1.79141920249979984539691017851, 2.7093158553591403892881438802, 3.06782494323683937096167358076, 4.22526246026030788792538716280, 4.9710243968635714651283129157, 5.5809559553159765801405447167, 6.21958355539166759758035205042, 7.07466546217238615664871485055, 7.62793145888900925906161704520, 8.3680228012223840590547328393, 9.02717197011898965156028418515, 9.29339306146937091997108017459, 10.262062896706204706175074103748, 11.359001806225048900898279024919, 12.034993830257402008673941201, 12.992602657608924938553326303363, 13.29014552730544984694607379032, 13.7029401549089705726941331916, 14.70740690097341198315468477371, 14.91834416639452356560804370416, 15.53110989415256461087787472348, 16.35562313754696328213421807521, 17.23597267649987586630423047133, 17.745869985339652871390900434596

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