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Properties

Label 1-4056-4056.3587-r0-0-0
Degree $1$
Conductor $4056$
Sign $-0.806 - 0.590i$
Analytic cond. $18.8359$
Root an. cond. $18.8359$
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  + (0.354 − 0.935i)5-s + (0.120 − 0.992i)7-s + (0.885 + 0.464i)11-s + (−0.120 + 0.992i)17-s − 19-s + 23-s + (−0.748 − 0.663i)25-s + (0.885 − 0.464i)29-s + (−0.748 + 0.663i)31-s + (−0.885 − 0.464i)35-s + (−0.748 + 0.663i)37-s + (−0.970 + 0.239i)41-s + (−0.748 − 0.663i)43-s + (−0.568 − 0.822i)47-s + (−0.970 − 0.239i)49-s + ⋯
L(s)  = 1  + (0.354 − 0.935i)5-s + (0.120 − 0.992i)7-s + (0.885 + 0.464i)11-s + (−0.120 + 0.992i)17-s − 19-s + 23-s + (−0.748 − 0.663i)25-s + (0.885 − 0.464i)29-s + (−0.748 + 0.663i)31-s + (−0.885 − 0.464i)35-s + (−0.748 + 0.663i)37-s + (−0.970 + 0.239i)41-s + (−0.748 − 0.663i)43-s + (−0.568 − 0.822i)47-s + (−0.970 − 0.239i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4056\)    =    \(2^{3} \cdot 3 \cdot 13^{2}\)
Sign: $-0.806 - 0.590i$
Analytic conductor: \(18.8359\)
Root analytic conductor: \(18.8359\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4056} (3587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4056,\ (0:\ ),\ -0.806 - 0.590i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3904742926 - 1.194198182i\)
\(L(\frac12)\) \(\approx\) \(0.3904742926 - 1.194198182i\)
\(L(1)\) \(\approx\) \(0.9980979648 - 0.3557147028i\)
\(L(1)\) \(\approx\) \(0.9980979648 - 0.3557147028i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.354 - 0.935i)T \)
7 \( 1 + (0.120 - 0.992i)T \)
11 \( 1 + (0.885 + 0.464i)T \)
17 \( 1 + (-0.120 + 0.992i)T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + (0.885 - 0.464i)T \)
31 \( 1 + (-0.748 + 0.663i)T \)
37 \( 1 + (-0.748 + 0.663i)T \)
41 \( 1 + (-0.970 + 0.239i)T \)
43 \( 1 + (-0.748 - 0.663i)T \)
47 \( 1 + (-0.568 - 0.822i)T \)
53 \( 1 + (0.120 - 0.992i)T \)
59 \( 1 + (-0.354 + 0.935i)T \)
61 \( 1 + (-0.120 - 0.992i)T \)
67 \( 1 + (-0.568 - 0.822i)T \)
71 \( 1 + (0.970 - 0.239i)T \)
73 \( 1 + (-0.885 - 0.464i)T \)
79 \( 1 + (-0.568 - 0.822i)T \)
83 \( 1 + (-0.970 - 0.239i)T \)
89 \( 1 + T \)
97 \( 1 + (0.354 + 0.935i)T \)
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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

−18.72325376417616327673116716685, −18.15772612630423724751681320225, −17.42319631784087454230771517604, −16.7979044125593386181621406568, −15.89737379229580560698088627710, −15.260877535785867142739076954956, −14.56637484665349202568959663396, −14.19045653792208543440219435025, −13.30010119937691649256347319644, −12.57384928010134459127334177283, −11.68284526006977864311603537571, −11.27723666990240255101449302998, −10.53942828938888274965398364386, −9.67364988473920953249662668885, −9.0012416284026169291603210, −8.50394080346759014553438627141, −7.39683031206218516037882447550, −6.72181302647311941651515591966, −6.156034874915247595188657960322, −5.402321430506943419063970777608, −4.58563547283205437318590205952, −3.51922418427098902244178811729, −2.86402764552842164044915892343, −2.17542601078800389613409482207, −1.26003871520646231877584564628, 0.333262993227194912501950728426, 1.56467321372733078450394934208, 1.741744200683694752379905185220, 3.24841583299985111748393097523, 4.015933033757630338678676608902, 4.6497603302351267223287421298, 5.26800328542435614077723836469, 6.478210486307376999622506838050, 6.72385051121533128079148431511, 7.80042801231057036612145698354, 8.59657231319788373502069301025, 8.990979824467169574318142348328, 10.09465332464149550525998798035, 10.355745047754988887480956749699, 11.357399103142545913497975057023, 12.113637054920168464639603045677, 12.79543395381378134617756556505, 13.35761546156153434384917261958, 14.007606795066236151699202027516, 14.82334874355813852512507818316, 15.3656774828540661495400771134, 16.46129074276485570523084705651, 16.92205685850923288801448789890, 17.29983649371075178725844079546, 17.928494628510087490320886790149

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