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Properties

Label 1-4008-4008.1325-r0-0-0
Degree $1$
Conductor $4008$
Sign $-0.968 + 0.248i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $-0.968 + 0.248i$
Analytic conductor: \(18.6130\)
Root analytic conductor: \(18.6130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (1325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (0:\ ),\ -0.968 + 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05957544103 - 0.4729348886i\)
\(L(\frac12)\) \(\approx\) \(-0.05957544103 - 0.4729348886i\)
\(L(1)\) \(\approx\) \(0.8343586981 - 0.2571789486i\)
\(L(1)\) \(\approx\) \(0.8343586981 - 0.2571789486i\)

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

−18.84024177751354631959587461685, −18.11536075858910270524608411547, −17.52411715612971515691141631089, −17.11571800918703731474848374420, −15.81262037087172493042817754661, −15.30234039469979570684439996187, −14.90573912757702075965941506275, −14.31078891924800825445517922628, −13.40203689878136366863036811332, −12.54857670555529578717313182793, −11.88167843852855274157418868050, −11.418174216027313830941645956004, −10.53659550014445407353547106821, −9.980308459169384343237381464655, −9.20959479140351096934573763794, −8.1873983374804381520272884564, −7.54855749138348413084586975419, −7.26983224917986949327278834781, −6.063424321569770198149732173584, −5.47144803731903025601362342122, −4.57664298889027815072840518349, −3.92311779976039231712464029884, −2.79079378571050659903709509531, −2.38917594342340968546310823792, −1.36009159214020742143212098181, 0.13402973956203479238553010704, 1.22997439328202020052830690583, 1.82308557122489429058406379122, 3.15912885109903953158048884360, 3.78928143836032344066095263333, 4.77771127047855089664365402546, 5.05339892328672623639320620815, 6.045791898804334947199852629423, 7.02909522206291364071327745519, 7.71242481096451541220671598785, 8.452670471970539884766331403743, 8.759458288339201546293730364576, 9.951246840884131075281387840408, 10.43827781769522646010165691732, 11.40950157267428115286241841213, 11.93491914562659852129947336582, 12.467655563651885160315501377555, 13.412335695888908129428736166961, 14.11559500442143762146310933098, 14.53460725484397327408317196795, 15.43469342022492630205809768795, 16.37872419577993232675382943431, 16.71939173407064798089004584279, 17.088762389781864658173283620135, 18.33136875022805277825569306246

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