Properties

Label 1-4008-4008.11-r0-0-0
Degree $1$
Conductor $4008$
Sign $0.153 - 0.988i$
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  + (0.489 + 0.872i)5-s + (−0.822 + 0.569i)7-s + (0.169 + 0.985i)11-s + (0.700 − 0.713i)13-s + (−0.929 + 0.369i)17-s + (0.206 − 0.978i)19-s + (−0.316 − 0.948i)23-s + (−0.521 + 0.853i)25-s + (−0.316 + 0.948i)29-s + (−0.421 − 0.906i)31-s + (−0.898 − 0.438i)35-s + (0.243 − 0.969i)37-s + (0.644 − 0.764i)41-s + (−0.455 − 0.890i)43-s + (−0.0944 + 0.995i)47-s + ⋯
L(s)  = 1  + (0.489 + 0.872i)5-s + (−0.822 + 0.569i)7-s + (0.169 + 0.985i)11-s + (0.700 − 0.713i)13-s + (−0.929 + 0.369i)17-s + (0.206 − 0.978i)19-s + (−0.316 − 0.948i)23-s + (−0.521 + 0.853i)25-s + (−0.316 + 0.948i)29-s + (−0.421 − 0.906i)31-s + (−0.898 − 0.438i)35-s + (0.243 − 0.969i)37-s + (0.644 − 0.764i)41-s + (−0.455 − 0.890i)43-s + (−0.0944 + 0.995i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.153 - 0.988i$
Analytic conductor: \(18.6130\)
Root analytic conductor: \(18.6130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (0:\ ),\ 0.153 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5604502758 - 0.4801819210i\)
\(L(\frac12)\) \(\approx\) \(0.5604502758 - 0.4801819210i\)
\(L(1)\) \(\approx\) \(0.9069607491 + 0.1158460429i\)
\(L(1)\) \(\approx\) \(0.9069607491 + 0.1158460429i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.489 + 0.872i)T \)
7 \( 1 + (-0.822 + 0.569i)T \)
11 \( 1 + (0.169 + 0.985i)T \)
13 \( 1 + (0.700 - 0.713i)T \)
17 \( 1 + (-0.929 + 0.369i)T \)
19 \( 1 + (0.206 - 0.978i)T \)
23 \( 1 + (-0.316 - 0.948i)T \)
29 \( 1 + (-0.316 + 0.948i)T \)
31 \( 1 + (-0.421 - 0.906i)T \)
37 \( 1 + (0.243 - 0.969i)T \)
41 \( 1 + (0.644 - 0.764i)T \)
43 \( 1 + (-0.455 - 0.890i)T \)
47 \( 1 + (-0.0944 + 0.995i)T \)
53 \( 1 + (-0.752 + 0.658i)T \)
59 \( 1 + (-0.929 - 0.369i)T \)
61 \( 1 + (-0.672 - 0.739i)T \)
67 \( 1 + (0.489 - 0.872i)T \)
71 \( 1 + (-0.843 - 0.537i)T \)
73 \( 1 + (0.997 + 0.0756i)T \)
79 \( 1 + (-0.974 + 0.225i)T \)
83 \( 1 + (0.0189 - 0.999i)T \)
89 \( 1 + (-0.988 + 0.150i)T \)
97 \( 1 + (0.421 - 0.906i)T \)
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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

−18.60728528181976589046132604425, −17.9622356517978889637309735366, −17.09168768530447330375376357911, −16.497270752905484566145039724277, −16.16931873941935472009059293922, −15.467417084734558318169095525412, −14.27714774389891890321074260654, −13.67276835756977973113009071804, −13.338841271842122833157028167840, −12.63487467804470939504347448155, −11.65923470672150152005686269241, −11.21999844511831706735242169274, −10.151620815993122217736129273886, −9.63310472880556585224483372852, −8.94083020032760065583741070518, −8.33306430757912915729450936103, −7.47285543038490142646233508674, −6.38935828855425724334916077692, −6.1354283831342920467350372074, −5.22549991440192538807411435732, −4.27085086719541831494419105385, −3.70177833843668531442856047596, −2.836938957755860487773263252328, −1.65440283899165742884069402180, −1.06604343387372346181050606644, 0.20877811055626074574137818575, 1.748751711165495823815212674742, 2.40376615596730388698076902837, 3.10128557942411445035763402487, 3.916263290027504799179265722745, 4.85240857961954820166310043816, 5.84413657779695603928171488050, 6.30556652290848761277775639276, 7.00772672201350079133452246138, 7.67495858689823721460401950992, 8.85085220481249210776390891332, 9.273182180562347731163821273080, 10.02827437629892255402250498850, 10.82438155053234179561612658084, 11.17757209097773892441355464059, 12.45146934462947776569146051284, 12.73878321174310628554936429346, 13.526580413009669047382422677332, 14.25853113653634204143385919240, 15.10558830360022019772691292907, 15.45468746625549569109915963107, 16.13052360380762343416659848331, 17.18438221578641099186945910404, 17.6950663331941738691671679995, 18.403690824859484109067902355052

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