Properties

Label 1-4024-4024.413-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.462 + 0.886i$
Analytic cond. $18.6873$
Root an. cond. $18.6873$
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  + (−0.968 − 0.247i)3-s + (0.986 + 0.161i)5-s + (0.143 + 0.989i)7-s + (0.877 + 0.479i)9-s + (−0.852 − 0.523i)11-s + (−0.241 − 0.970i)13-s + (−0.915 − 0.401i)15-s + (−0.905 + 0.424i)17-s + (0.649 + 0.760i)19-s + (0.106 − 0.994i)21-s + (−0.180 − 0.983i)23-s + (0.947 + 0.319i)25-s + (−0.731 − 0.682i)27-s + (0.463 − 0.886i)29-s + (−0.325 + 0.945i)31-s + ⋯
L(s)  = 1  + (−0.968 − 0.247i)3-s + (0.986 + 0.161i)5-s + (0.143 + 0.989i)7-s + (0.877 + 0.479i)9-s + (−0.852 − 0.523i)11-s + (−0.241 − 0.970i)13-s + (−0.915 − 0.401i)15-s + (−0.905 + 0.424i)17-s + (0.649 + 0.760i)19-s + (0.106 − 0.994i)21-s + (−0.180 − 0.983i)23-s + (0.947 + 0.319i)25-s + (−0.731 − 0.682i)27-s + (0.463 − 0.886i)29-s + (−0.325 + 0.945i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.462 + 0.886i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.462 + 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9649682391 + 0.5853641120i\)
\(L(\frac12)\) \(\approx\) \(0.9649682391 + 0.5853641120i\)
\(L(1)\) \(\approx\) \(0.8641167905 + 0.07562819286i\)
\(L(1)\) \(\approx\) \(0.8641167905 + 0.07562819286i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.968 - 0.247i)T \)
5 \( 1 + (0.986 + 0.161i)T \)
7 \( 1 + (0.143 + 0.989i)T \)
11 \( 1 + (-0.852 - 0.523i)T \)
13 \( 1 + (-0.241 - 0.970i)T \)
17 \( 1 + (-0.905 + 0.424i)T \)
19 \( 1 + (0.649 + 0.760i)T \)
23 \( 1 + (-0.180 - 0.983i)T \)
29 \( 1 + (0.463 - 0.886i)T \)
31 \( 1 + (-0.325 + 0.945i)T \)
37 \( 1 + (-0.780 - 0.625i)T \)
41 \( 1 + (-0.858 - 0.512i)T \)
43 \( 1 + (0.253 + 0.967i)T \)
47 \( 1 + (-0.686 + 0.726i)T \)
53 \( 1 + (0.649 - 0.760i)T \)
59 \( 1 + (0.772 + 0.635i)T \)
61 \( 1 + (-0.695 + 0.718i)T \)
67 \( 1 + (0.915 - 0.401i)T \)
71 \( 1 + (0.962 + 0.271i)T \)
73 \( 1 + (-0.977 + 0.211i)T \)
79 \( 1 + (0.939 + 0.343i)T \)
83 \( 1 + (0.0312 - 0.999i)T \)
89 \( 1 + (0.980 + 0.198i)T \)
97 \( 1 + (0.217 - 0.976i)T \)
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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

−18.093394063455336899263516333458, −17.62535108032210566180771956938, −17.059435649739229367986836860569, −16.48881524637103650617453845262, −15.76705958532908809141675459243, −15.11307100736846984736683643283, −14.04119350030938124113658281673, −13.504340508012936876362340059168, −13.04482254368220698983890599674, −12.090896011015044837712263367925, −11.38076339528694059224171749123, −10.76627825239984450692916768515, −10.0445856204013730895243514188, −9.6046747962397245979925988589, −8.85222622861824704408008272357, −7.617068557991418413985252302749, −6.8842717124512875218188738096, −6.58428983823230988415055656812, −5.36371105566146037817697851712, −5.00785775236697461710953884461, −4.358594126043314210151157038525, −3.36782725188611717066844355940, −2.15887577753211772002711026944, −1.51996623643922286001935298704, −0.437760646502472552387251554431, 0.84018697511175973463443516226, 1.941555331503042785905624611973, 2.4645944343729670404464760261, 3.408862979679848540403455722045, 4.79566625762561360561615700214, 5.242919592930811126192320356661, 5.94443329965623570701309999491, 6.31021352916970503246461987235, 7.29223759493616840151680944749, 8.19220023544680773485035331513, 8.78809414241215845566027479537, 9.87439695318258815563296672053, 10.35809348734341492383478851868, 10.92692762038423795559512677958, 11.76104789017284163095320770571, 12.55788496299221485645637995589, 12.92723011853319296004991040032, 13.68363249249783086133441724269, 14.49692322811053464204559180254, 15.32732754312929420149881936573, 15.97274805989795514697981427022, 16.534271713583793981695200070829, 17.60964314781972134969612827180, 17.76158359357603487217368676516, 18.41802011051707305160470818177

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