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Properties

Label 1-4008-4008.437-r0-0-0
Degree $1$
Conductor $4008$
Sign $0.695 - 0.718i$
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  + (0.942 − 0.334i)5-s + (−0.800 − 0.599i)7-s + (−0.993 + 0.113i)11-s + (0.862 − 0.505i)13-s + (−0.700 − 0.713i)17-s + (−0.614 + 0.788i)19-s + (0.672 + 0.739i)23-s + (0.776 − 0.629i)25-s + (0.672 − 0.739i)29-s + (0.726 + 0.686i)31-s + (−0.954 − 0.298i)35-s + (0.351 + 0.936i)37-s + (0.0567 + 0.998i)41-s + (0.206 − 0.978i)43-s + (−0.553 + 0.832i)47-s + ⋯
L(s)  = 1  + (0.942 − 0.334i)5-s + (−0.800 − 0.599i)7-s + (−0.993 + 0.113i)11-s + (0.862 − 0.505i)13-s + (−0.700 − 0.713i)17-s + (−0.614 + 0.788i)19-s + (0.672 + 0.739i)23-s + (0.776 − 0.629i)25-s + (0.672 − 0.739i)29-s + (0.726 + 0.686i)31-s + (−0.954 − 0.298i)35-s + (0.351 + 0.936i)37-s + (0.0567 + 0.998i)41-s + (0.206 − 0.978i)43-s + (−0.553 + 0.832i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.695 - 0.718i$
Analytic conductor: \(18.6130\)
Root analytic conductor: \(18.6130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (0:\ ),\ 0.695 - 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.627456958 - 0.6897403128i\)
\(L(\frac12)\) \(\approx\) \(1.627456958 - 0.6897403128i\)
\(L(1)\) \(\approx\) \(1.124022818 - 0.1813090633i\)
\(L(1)\) \(\approx\) \(1.124022818 - 0.1813090633i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.942 - 0.334i)T \)
7 \( 1 + (-0.800 - 0.599i)T \)
11 \( 1 + (-0.993 + 0.113i)T \)
13 \( 1 + (0.862 - 0.505i)T \)
17 \( 1 + (-0.700 - 0.713i)T \)
19 \( 1 + (-0.614 + 0.788i)T \)
23 \( 1 + (0.672 + 0.739i)T \)
29 \( 1 + (0.672 - 0.739i)T \)
31 \( 1 + (0.726 + 0.686i)T \)
37 \( 1 + (0.351 + 0.936i)T \)
41 \( 1 + (0.0567 + 0.998i)T \)
43 \( 1 + (0.206 - 0.978i)T \)
47 \( 1 + (-0.553 + 0.832i)T \)
53 \( 1 + (0.843 + 0.537i)T \)
59 \( 1 + (0.700 - 0.713i)T \)
61 \( 1 + (0.881 - 0.472i)T \)
67 \( 1 + (-0.942 - 0.334i)T \)
71 \( 1 + (0.929 + 0.369i)T \)
73 \( 1 + (0.455 + 0.890i)T \)
79 \( 1 + (-0.988 + 0.150i)T \)
83 \( 1 + (-0.489 - 0.872i)T \)
89 \( 1 + (0.584 + 0.811i)T \)
97 \( 1 + (0.726 - 0.686i)T \)
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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

−18.4558020462441083519984804916, −18.013035742933581676955254795195, −17.29951709068780376581127560085, −16.437752284926655696581456697744, −15.90801309385241320424755430355, −15.145079161363230663871977739914, −14.57599939990647278917599335274, −13.525875924448190980228703264482, −13.16833667866737545115690267740, −12.69673809860855868428076521952, −11.636412164392696830622034521384, −10.73893908601535707114525868968, −10.45064837852333837535570588620, −9.521018345778469002257264607413, −8.80511724120657492977002165974, −8.424139868229292133525916322370, −7.08984288401547610416205043625, −6.5309332695723536497038815301, −5.97695528781198261596847824145, −5.24462946216728953853520807694, −4.3451404866909734780967775021, −3.34825618255195459343030225477, −2.47509966429940105482887015888, −2.152097461316926055211106157315, −0.8097751483974239259427215302, 0.63900471040375814270354933181, 1.49549041567474848128246021995, 2.59062214342127147737395068696, 3.112003723064374104160635239834, 4.17971097797729358879687374676, 4.94964483163461499063414082365, 5.7311443139547747940150036202, 6.39257719670599092548447881795, 7.02120337758315081918501927275, 8.05688365039884260141750421624, 8.61103923618094992485138809265, 9.57644874011232237392465471118, 10.05932968715565908137615859059, 10.62999834599407068082483283200, 11.41331557302770456264028309182, 12.50886222681997751454761839930, 13.060584830831526919388723608149, 13.5272724995854520603339843332, 14.01348622759576318744382010122, 15.12046914619403462198280175218, 15.83280209306719658641578284209, 16.25850432353734722374102686267, 17.185027711450700471872417964516, 17.58239682499451439919526769267, 18.39310712709066246882219960587

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