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Properties

Label 1-4008-4008.53-r0-0-0
Degree $1$
Conductor $4008$
Sign $0.138 - 0.990i$
Analytic cond. $18.6130$
Root an. cond. $18.6130$
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  + (−0.553 − 0.832i)5-s + (−0.993 + 0.113i)7-s + (−0.752 + 0.658i)11-s + (0.672 + 0.739i)13-s + (−0.316 + 0.948i)17-s + (−0.862 − 0.505i)19-s + (−0.999 − 0.0378i)23-s + (−0.387 + 0.922i)25-s + (−0.999 + 0.0378i)29-s + (0.822 + 0.569i)31-s + (0.644 + 0.764i)35-s + (−0.942 − 0.334i)37-s + (0.351 + 0.936i)41-s + (−0.700 − 0.713i)43-s + (0.455 + 0.890i)47-s + ⋯
L(s)  = 1  + (−0.553 − 0.832i)5-s + (−0.993 + 0.113i)7-s + (−0.752 + 0.658i)11-s + (0.672 + 0.739i)13-s + (−0.316 + 0.948i)17-s + (−0.862 − 0.505i)19-s + (−0.999 − 0.0378i)23-s + (−0.387 + 0.922i)25-s + (−0.999 + 0.0378i)29-s + (0.822 + 0.569i)31-s + (0.644 + 0.764i)35-s + (−0.942 − 0.334i)37-s + (0.351 + 0.936i)41-s + (−0.700 − 0.713i)43-s + (0.455 + 0.890i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.138 - 0.990i$
Analytic conductor: \(18.6130\)
Root analytic conductor: \(18.6130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (0:\ ),\ 0.138 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3374896372 - 0.2935363605i\)
\(L(\frac12)\) \(\approx\) \(0.3374896372 - 0.2935363605i\)
\(L(1)\) \(\approx\) \(0.6748623531 + 0.01324669192i\)
\(L(1)\) \(\approx\) \(0.6748623531 + 0.01324669192i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.553 - 0.832i)T \)
7 \( 1 + (-0.993 + 0.113i)T \)
11 \( 1 + (-0.752 + 0.658i)T \)
13 \( 1 + (0.672 + 0.739i)T \)
17 \( 1 + (-0.316 + 0.948i)T \)
19 \( 1 + (-0.862 - 0.505i)T \)
23 \( 1 + (-0.999 - 0.0378i)T \)
29 \( 1 + (-0.999 + 0.0378i)T \)
31 \( 1 + (0.822 + 0.569i)T \)
37 \( 1 + (-0.942 - 0.334i)T \)
41 \( 1 + (0.351 + 0.936i)T \)
43 \( 1 + (-0.700 - 0.713i)T \)
47 \( 1 + (0.455 + 0.890i)T \)
53 \( 1 + (-0.898 - 0.438i)T \)
59 \( 1 + (0.316 + 0.948i)T \)
61 \( 1 + (0.521 - 0.853i)T \)
67 \( 1 + (0.553 - 0.832i)T \)
71 \( 1 + (0.954 + 0.298i)T \)
73 \( 1 + (-0.929 + 0.369i)T \)
79 \( 1 + (-0.421 - 0.906i)T \)
83 \( 1 + (0.0944 + 0.995i)T \)
89 \( 1 + (-0.726 - 0.686i)T \)
97 \( 1 + (0.822 - 0.569i)T \)
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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

−18.79368309635879682450156436979, −18.15316246780264533126005933837, −17.30401625090415788455091981111, −16.40549599431049022019402801832, −15.74707225545226226988449315932, −15.531017750417029570013124271139, −14.571022372754578009630589472549, −13.75020308547726808782715357260, −13.271262345268686240391446526190, −12.51029356418436954440740758837, −11.686836240260761330549900962942, −11.013045579208041431941313892848, −10.33516630746511846633457262441, −9.88957421411024266382289210782, −8.80021149887428998135693039050, −8.11316294993395056548870693215, −7.47182343303634963366506894728, −6.628785616532392293173095253411, −6.06307521124378087588918853364, −5.338051889320533408080627429936, −4.10667059269565543340695131440, −3.54559609225135610342367354608, −2.87295496941778265459783236875, −2.14709118877074134112122328709, −0.60742168620493573517049805326, 0.19842149028839051299251725250, 1.548363502151051932291845408428, 2.25819330481162475063507457908, 3.38774974657371815837669810793, 4.07134156136830133148227922073, 4.66821645759024109967405245407, 5.62745342374952775141682759173, 6.37487207771873706296139697487, 7.04390990773697344431752092906, 7.99287559678881339698192370665, 8.58546396427719177968118285251, 9.229555472734930454368540305483, 9.98959984341158384649003065, 10.71785759777469522367874040827, 11.519986925206849878204881859831, 12.33953270911750137377169303136, 12.84592968838602623351463307893, 13.28331719599095412037186852587, 14.18423739498175058023343215362, 15.22072373763144760118788496586, 15.681997857295377291478969916764, 16.15074427153099743881280043710, 16.9447493403437696487775611181, 17.49789391866413492263412352654, 18.46613500612914732531769360997

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