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Properties

Label 1-4000-4000.147-r0-0-0
Degree $1$
Conductor $4000$
Sign $0.895 - 0.444i$
Analytic cond. $18.5759$
Root an. cond. $18.5759$
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  + (−0.995 − 0.0941i)3-s + (−0.309 − 0.951i)7-s + (0.982 + 0.187i)9-s + (−0.790 + 0.612i)11-s + (0.562 + 0.827i)13-s + (−0.248 − 0.968i)17-s + (−0.995 + 0.0941i)19-s + (0.218 + 0.975i)21-s + (−0.728 − 0.684i)23-s + (−0.960 − 0.278i)27-s + (−0.661 + 0.750i)29-s + (−0.968 + 0.248i)31-s + (0.844 − 0.535i)33-s + (0.278 + 0.960i)37-s + (−0.481 − 0.876i)39-s + ⋯
L(s)  = 1  + (−0.995 − 0.0941i)3-s + (−0.309 − 0.951i)7-s + (0.982 + 0.187i)9-s + (−0.790 + 0.612i)11-s + (0.562 + 0.827i)13-s + (−0.248 − 0.968i)17-s + (−0.995 + 0.0941i)19-s + (0.218 + 0.975i)21-s + (−0.728 − 0.684i)23-s + (−0.960 − 0.278i)27-s + (−0.661 + 0.750i)29-s + (−0.968 + 0.248i)31-s + (0.844 − 0.535i)33-s + (0.278 + 0.960i)37-s + (−0.481 − 0.876i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.895 - 0.444i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ 0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6893968891 - 0.1615061796i\)
\(L(\frac12)\) \(\approx\) \(0.6893968891 - 0.1615061796i\)
\(L(1)\) \(\approx\) \(0.6527036477 - 0.05465995635i\)
\(L(1)\) \(\approx\) \(0.6527036477 - 0.05465995635i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.995 - 0.0941i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-0.790 + 0.612i)T \)
13 \( 1 + (0.562 + 0.827i)T \)
17 \( 1 + (-0.248 - 0.968i)T \)
19 \( 1 + (-0.995 + 0.0941i)T \)
23 \( 1 + (-0.728 - 0.684i)T \)
29 \( 1 + (-0.661 + 0.750i)T \)
31 \( 1 + (-0.968 + 0.248i)T \)
37 \( 1 + (0.278 + 0.960i)T \)
41 \( 1 + (0.684 + 0.728i)T \)
43 \( 1 + (0.987 + 0.156i)T \)
47 \( 1 + (-0.368 - 0.929i)T \)
53 \( 1 + (0.975 - 0.218i)T \)
59 \( 1 + (-0.940 - 0.338i)T \)
61 \( 1 + (-0.0314 - 0.999i)T \)
67 \( 1 + (0.661 + 0.750i)T \)
71 \( 1 + (-0.368 - 0.929i)T \)
73 \( 1 + (-0.425 + 0.904i)T \)
79 \( 1 + (0.637 + 0.770i)T \)
83 \( 1 + (-0.0941 - 0.995i)T \)
89 \( 1 + (-0.904 - 0.425i)T \)
97 \( 1 + (-0.998 + 0.0627i)T \)
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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

−18.27880062478582045665673861078, −18.00076974355488751493406771531, −17.18552168355259597401252523030, −16.49029361672613525609616381930, −15.68305932778185992152168179964, −15.47925504059721352557346011802, −14.65912335090053214737075667530, −13.47433032190085228203587641212, −12.85059016737163494628053588584, −12.52076557361487356924290413776, −11.59257693266397997320368219497, −10.828385371669814690672962249209, −10.572697797335549822088229779934, −9.55994872495529106122025148209, −8.85460037463839305989464832588, −8.02544020394529828365574526188, −7.355085627189748074636393884027, −6.07373887148368164144786794470, −5.93920748814490585807467999808, −5.3919880780852522936864077473, −4.24104301619213580959048216469, −3.63221559284597197770503979335, −2.52552746473384769084434474846, −1.76910580458723834353638865915, −0.50669188381786889503277755688, 0.441341976858607516205639186196, 1.53244752660456788678469694443, 2.31473648386601981515192974499, 3.54998789706456482376368738636, 4.3782590265904193911619864744, 4.79515177300235635221595209008, 5.791891601724102677179654211901, 6.545642099231593648789482375176, 7.086019485464828339407979016984, 7.70368691666204988150808382769, 8.72051858795679508669268330697, 9.69178608872300450678383484791, 10.1962794614437631452477436104, 10.998605532051955356132259460248, 11.322108722669085833309879801921, 12.40040712589807423275087106438, 12.85241290603688473696667404918, 13.52016526204985327039986656846, 14.25902668492936769455244610642, 15.14445658255172718329514428836, 16.04083815391545612936296637257, 16.40771108338189718567110687127, 16.9616588726995974960856710748, 17.80934751723243375284621949603, 18.363619607669263553433677648823

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