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Properties

Label 1-4056-4056.2621-r0-0-0
Degree $1$
Conductor $4056$
Sign $-0.529 - 0.848i$
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  + (0.822 + 0.568i)5-s + (−0.663 + 0.748i)7-s + (0.239 + 0.970i)11-s + (−0.748 − 0.663i)17-s − i·19-s + 23-s + (0.354 + 0.935i)25-s + (−0.970 − 0.239i)29-s + (−0.935 − 0.354i)31-s + (−0.970 + 0.239i)35-s + (−0.935 − 0.354i)37-s + (−0.992 − 0.120i)41-s + (−0.354 − 0.935i)43-s + (0.464 + 0.885i)47-s + (−0.120 − 0.992i)49-s + ⋯
L(s)  = 1  + (0.822 + 0.568i)5-s + (−0.663 + 0.748i)7-s + (0.239 + 0.970i)11-s + (−0.748 − 0.663i)17-s − i·19-s + 23-s + (0.354 + 0.935i)25-s + (−0.970 − 0.239i)29-s + (−0.935 − 0.354i)31-s + (−0.970 + 0.239i)35-s + (−0.935 − 0.354i)37-s + (−0.992 − 0.120i)41-s + (−0.354 − 0.935i)43-s + (0.464 + 0.885i)47-s + (−0.120 − 0.992i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.529 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.529 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4056\)    =    \(2^{3} \cdot 3 \cdot 13^{2}\)
Sign: $-0.529 - 0.848i$
Analytic conductor: \(18.8359\)
Root analytic conductor: \(18.8359\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4056} (2621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4056,\ (0:\ ),\ -0.529 - 0.848i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1302949872 - 0.2348295495i\)
\(L(\frac12)\) \(\approx\) \(0.1302949872 - 0.2348295495i\)
\(L(1)\) \(\approx\) \(0.9062593573 + 0.1452990605i\)
\(L(1)\) \(\approx\) \(0.9062593573 + 0.1452990605i\)

Euler product

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

−18.71486427732324457312891740758, −17.97429657139498138041110379553, −17.02492744593606181962135776697, −16.79331355596711773299085603302, −16.24399691512221425453248345881, −15.308481920719240939535680074484, −14.46557991917286805503929304450, −13.81257835902681265997258546148, −13.17213525679499027828496686056, −12.81962437744301152992106339769, −11.89495439912304946229056059109, −10.92723282356687678810415649182, −10.44107793990118292920651754264, −9.70936750372717269389415951444, −8.90898581908965151627286698708, −8.51353028145499927705558234622, −7.4240689918868935174624540940, −6.66778494487431747112363462570, −6.00175698811486192534645651430, −5.39128455003040156724638664597, −4.43921958336492101583763642799, −3.62891494259868001519063740812, −2.986820324900365900137430014410, −1.750244421355984369129601950941, −1.206441450240555785975371829109, 0.06901283994184399482796565738, 1.66560047272483790539156487458, 2.282186108770002473818755073572, 2.95908091865427220179225951151, 3.8100279566991979570228082037, 5.00276080681750467832793472884, 5.37954382130370442174798027437, 6.475498070204494278277838064842, 6.83583446511687816651414128199, 7.51085461586244928568061939399, 8.84238047992883426734701967766, 9.28520768824335633242458648881, 9.72424027777298406604703589735, 10.69407884098916605312941142394, 11.26007026364625151699763501205, 12.1168407656877315373395448770, 12.909375989743065588611505190635, 13.33603016210948905167316048490, 14.135893638938941871495049061977, 15.02025464543736061791868582506, 15.31915164637278131068195966440, 16.10444263677752918645009186547, 17.13689814649118573898780019173, 17.46975309532013624878194606040, 18.30216176586169555489921615886

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