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Properties

Label 1-4000-4000.61-r0-0-0
Degree $1$
Conductor $4000$
Sign $0.0800 - 0.996i$
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  + (0.960 + 0.278i)3-s + (−0.587 + 0.809i)7-s + (0.844 + 0.535i)9-s + (−0.917 + 0.397i)11-s + (−0.975 + 0.218i)13-s + (−0.728 + 0.684i)17-s + (−0.278 − 0.960i)19-s + (−0.790 + 0.612i)21-s + (−0.770 − 0.637i)23-s + (0.661 + 0.750i)27-s + (0.562 − 0.827i)29-s + (0.728 − 0.684i)31-s + (−0.992 + 0.125i)33-s + (−0.750 − 0.661i)37-s + (−0.998 − 0.0627i)39-s + ⋯
L(s)  = 1  + (0.960 + 0.278i)3-s + (−0.587 + 0.809i)7-s + (0.844 + 0.535i)9-s + (−0.917 + 0.397i)11-s + (−0.975 + 0.218i)13-s + (−0.728 + 0.684i)17-s + (−0.278 − 0.960i)19-s + (−0.790 + 0.612i)21-s + (−0.770 − 0.637i)23-s + (0.661 + 0.750i)27-s + (0.562 − 0.827i)29-s + (0.728 − 0.684i)31-s + (−0.992 + 0.125i)33-s + (−0.750 − 0.661i)37-s + (−0.998 − 0.0627i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0800 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0800 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.0800 - 0.996i$
Analytic conductor: \(18.5759\)
Root analytic conductor: \(18.5759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4000,\ (0:\ ),\ 0.0800 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5701367648 - 0.5261991915i\)
\(L(\frac12)\) \(\approx\) \(0.5701367648 - 0.5261991915i\)
\(L(1)\) \(\approx\) \(1.036603841 + 0.1429604807i\)
\(L(1)\) \(\approx\) \(1.036603841 + 0.1429604807i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.960 + 0.278i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (-0.917 + 0.397i)T \)
13 \( 1 + (-0.975 + 0.218i)T \)
17 \( 1 + (-0.728 + 0.684i)T \)
19 \( 1 + (-0.278 - 0.960i)T \)
23 \( 1 + (-0.770 - 0.637i)T \)
29 \( 1 + (0.562 - 0.827i)T \)
31 \( 1 + (0.728 - 0.684i)T \)
37 \( 1 + (-0.750 - 0.661i)T \)
41 \( 1 + (0.770 - 0.637i)T \)
43 \( 1 + (0.891 + 0.453i)T \)
47 \( 1 + (0.425 - 0.904i)T \)
53 \( 1 + (-0.790 + 0.612i)T \)
59 \( 1 + (-0.860 + 0.509i)T \)
61 \( 1 + (-0.995 + 0.0941i)T \)
67 \( 1 + (-0.827 + 0.562i)T \)
71 \( 1 + (0.904 + 0.425i)T \)
73 \( 1 + (0.248 + 0.968i)T \)
79 \( 1 + (-0.876 + 0.481i)T \)
83 \( 1 + (-0.278 - 0.960i)T \)
89 \( 1 + (-0.248 - 0.968i)T \)
97 \( 1 + (-0.187 - 0.982i)T \)
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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

−18.75745138342459320640463452555, −18.02428649041017433501631560611, −17.41116087730456119487277816163, −16.46939086840650241212211764066, −15.80462336226832575647616737120, −15.37575310912898181476544582118, −14.190598624731588297731890056796, −14.04275340200799834224711671262, −13.26860115261573804399684579846, −12.57664864331083337190319688446, −12.07649154435898393778238706198, −10.80972471367148256705211336911, −10.29648672660252650126986251181, −9.61755231820073253485832890285, −8.97307712337998637419512562926, −7.88860634993868913019059020036, −7.72982288090380960800205696866, −6.80893673251478690523532003946, −6.17201287728323310532384154325, −4.99869688349020931666129533956, −4.34063690375717755096758676419, −3.34725923592654094616691489967, −2.900129562586021659368795662884, −2.02959523123984453909003335358, −1.02421945561186037932259519928, 0.18674974989113249174458940704, 1.92535355471164546123663837261, 2.510635886413485491351590168045, 2.86612016706462619335351261118, 4.19949489288038902452451029905, 4.52623253683718204429546302486, 5.55943956954893437640997914358, 6.39317580859249911294632712493, 7.249626677339147599006397278728, 7.89849725050732502194248295671, 8.72649773937692708181811548189, 9.17885117718139323222426102395, 10.03373578356411526689789442545, 10.41975082097798146473679236035, 11.451998408084697625790621722117, 12.51498008436871321584175943137, 12.719009969444151056044250042778, 13.6056015323798347837082501377, 14.24288690016331136130179103878, 15.11649405456275056192580186641, 15.5535363814212012534039622895, 15.90188359928230052935465199909, 16.99039447699637460938526625021, 17.67836917621428395113813191762, 18.50314277798725621621693037643

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