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Properties

Label 1-4001-4001.155-r0-0-0
Degree $1$
Conductor $4001$
Sign $0.646 - 0.762i$
Analytic cond. $18.5805$
Root an. cond. $18.5805$
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  + (0.702 + 0.711i)2-s + (−0.998 + 0.0596i)3-s + (−0.0125 + 0.999i)4-s + (0.248 − 0.968i)5-s + (−0.743 − 0.668i)6-s + (0.503 − 0.863i)7-s + (−0.720 + 0.693i)8-s + (0.992 − 0.119i)9-s + (0.863 − 0.503i)10-s + (0.972 − 0.233i)11-s + (−0.0471 − 0.998i)12-s + (−0.934 + 0.356i)13-s + (0.968 − 0.248i)14-s + (−0.190 + 0.981i)15-s + (−0.999 − 0.0251i)16-s + (0.828 − 0.559i)17-s + ⋯
L(s)  = 1  + (0.702 + 0.711i)2-s + (−0.998 + 0.0596i)3-s + (−0.0125 + 0.999i)4-s + (0.248 − 0.968i)5-s + (−0.743 − 0.668i)6-s + (0.503 − 0.863i)7-s + (−0.720 + 0.693i)8-s + (0.992 − 0.119i)9-s + (0.863 − 0.503i)10-s + (0.972 − 0.233i)11-s + (−0.0471 − 0.998i)12-s + (−0.934 + 0.356i)13-s + (0.968 − 0.248i)14-s + (−0.190 + 0.981i)15-s + (−0.999 − 0.0251i)16-s + (0.828 − 0.559i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4001\)
Sign: $0.646 - 0.762i$
Analytic conductor: \(18.5805\)
Root analytic conductor: \(18.5805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4001} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4001,\ (0:\ ),\ 0.646 - 0.762i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.600952915 - 0.7416898507i\)
\(L(\frac12)\) \(\approx\) \(1.600952915 - 0.7416898507i\)
\(L(1)\) \(\approx\) \(1.213177465 + 0.09861179634i\)
\(L(1)\) \(\approx\) \(1.213177465 + 0.09861179634i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4001 \( 1 \)
good2 \( 1 + (0.702 + 0.711i)T \)
3 \( 1 + (-0.998 + 0.0596i)T \)
5 \( 1 + (0.248 - 0.968i)T \)
7 \( 1 + (0.503 - 0.863i)T \)
11 \( 1 + (0.972 - 0.233i)T \)
13 \( 1 + (-0.934 + 0.356i)T \)
17 \( 1 + (0.828 - 0.559i)T \)
19 \( 1 + (-0.175 - 0.984i)T \)
23 \( 1 + (-0.959 + 0.282i)T \)
29 \( 1 + (0.902 - 0.431i)T \)
31 \( 1 + (-0.902 - 0.431i)T \)
37 \( 1 + (0.913 + 0.405i)T \)
41 \( 1 + (0.764 + 0.644i)T \)
43 \( 1 + (0.918 + 0.394i)T \)
47 \( 1 + (0.879 - 0.476i)T \)
53 \( 1 + (-0.999 + 0.0157i)T \)
59 \( 1 + (0.338 - 0.940i)T \)
61 \( 1 + (0.999 + 0.00628i)T \)
67 \( 1 + (0.371 + 0.928i)T \)
71 \( 1 + (-0.675 + 0.737i)T \)
73 \( 1 + (0.263 + 0.964i)T \)
79 \( 1 + (-0.368 - 0.929i)T \)
83 \( 1 + (0.847 - 0.530i)T \)
89 \( 1 + (-0.992 + 0.119i)T \)
97 \( 1 + (0.0439 - 0.999i)T \)
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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

−18.67578376489535878678571397047, −17.877038145626960428122881997887, −17.55673728765267549665019399912, −16.52131353418944667731537680094, −15.72553396161092655056676299226, −14.85161305720075595999120204944, −14.521189331030095158521387043960, −13.98052787336676565477824876184, −12.69885991465749306431003231595, −12.17523469858937802042628167477, −12.01685306344552367698770948133, −10.9512514433549462577873346439, −10.60484959033915829722540538164, −9.82904843991288666497136047233, −9.27642201491079261882183904368, −7.93595319190225219305042403455, −7.131156189578684641706486023, −6.226126166887264737198910132821, −5.85172147274470605069404163802, −5.21767804839632938254844744657, −4.25770832874516800835437791280, −3.648128478821770577313956093982, −2.53794210929281088623591456933, −1.93097359668659574910864411485, −1.124591655133235853079492021449, 0.48194498052450072735093907155, 1.348891510770006520993381761, 2.46801758685773335519897852166, 3.867139706872512791630826392953, 4.36572659054115656973674239642, 4.86076212101499883211724051532, 5.6202068642557767721124691052, 6.26572018959198723493377321584, 7.104457600733124642491849343, 7.6138395868419221997002547666, 8.47012276860763780918892785768, 9.50337187792703177357974785520, 9.88613296054657746108062783058, 11.27779242989527548680749847371, 11.54955966795234713434234390194, 12.327949263226752874779118322634, 12.86427568857630587312028237703, 13.6938438744914649589548039334, 14.23257695082870258271291002002, 14.923955159209662457694963380752, 16.07642978180330554942554294223, 16.26921047028707775490863203956, 17.03086130185879618213287783073, 17.43157580904927903615406354085, 17.81983008926292596625398729738

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