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Properties

Label 1-4008-4008.1379-r1-0-0
Degree $1$
Conductor $4008$
Sign $0.997 - 0.0707i$
Analytic cond. $430.719$
Root an. cond. $430.719$
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  + (−0.988 − 0.150i)5-s + (−0.553 + 0.832i)7-s + (0.455 + 0.890i)11-s + (−0.993 + 0.113i)13-s + (−0.169 + 0.985i)17-s + (−0.800 + 0.599i)19-s + (0.752 − 0.658i)23-s + (0.954 + 0.298i)25-s + (−0.752 − 0.658i)29-s + (−0.489 − 0.872i)31-s + (0.672 − 0.739i)35-s + (−0.982 + 0.188i)37-s + (−0.521 + 0.853i)41-s + (−0.822 − 0.569i)43-s + (0.421 + 0.906i)47-s + ⋯
L(s)  = 1  + (−0.988 − 0.150i)5-s + (−0.553 + 0.832i)7-s + (0.455 + 0.890i)11-s + (−0.993 + 0.113i)13-s + (−0.169 + 0.985i)17-s + (−0.800 + 0.599i)19-s + (0.752 − 0.658i)23-s + (0.954 + 0.298i)25-s + (−0.752 − 0.658i)29-s + (−0.489 − 0.872i)31-s + (0.672 − 0.739i)35-s + (−0.982 + 0.188i)37-s + (−0.521 + 0.853i)41-s + (−0.822 − 0.569i)43-s + (0.421 + 0.906i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.997 - 0.0707i$
Analytic conductor: \(430.719\)
Root analytic conductor: \(430.719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (1379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (1:\ ),\ 0.997 - 0.0707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3582018832 + 0.01268579751i\)
\(L(\frac12)\) \(\approx\) \(0.3582018832 + 0.01268579751i\)
\(L(1)\) \(\approx\) \(0.6273493640 + 0.1299464227i\)
\(L(1)\) \(\approx\) \(0.6273493640 + 0.1299464227i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.988 - 0.150i)T \)
7 \( 1 + (-0.553 + 0.832i)T \)
11 \( 1 + (0.455 + 0.890i)T \)
13 \( 1 + (-0.993 + 0.113i)T \)
17 \( 1 + (-0.169 + 0.985i)T \)
19 \( 1 + (-0.800 + 0.599i)T \)
23 \( 1 + (0.752 - 0.658i)T \)
29 \( 1 + (-0.752 - 0.658i)T \)
31 \( 1 + (-0.489 - 0.872i)T \)
37 \( 1 + (-0.982 + 0.188i)T \)
41 \( 1 + (-0.521 + 0.853i)T \)
43 \( 1 + (-0.822 - 0.569i)T \)
47 \( 1 + (0.421 + 0.906i)T \)
53 \( 1 + (0.700 + 0.713i)T \)
59 \( 1 + (-0.169 - 0.985i)T \)
61 \( 1 + (0.843 + 0.537i)T \)
67 \( 1 + (-0.988 + 0.150i)T \)
71 \( 1 + (-0.862 - 0.505i)T \)
73 \( 1 + (-0.614 - 0.788i)T \)
79 \( 1 + (-0.914 - 0.404i)T \)
83 \( 1 + (0.974 + 0.225i)T \)
89 \( 1 + (0.243 + 0.969i)T \)
97 \( 1 + (0.489 - 0.872i)T \)
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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

−18.50955985727549718969302296993, −17.46224993787440250150420966929, −16.90870147348569723604035837209, −16.26395936550353480245040850987, −15.72370705514646758562029093610, −14.85659889069298396081507999700, −14.34698420057807437595041438229, −13.44573857922295939506573457567, −12.944521157713413656167721318958, −11.993679664181904016271023104950, −11.50282438499232292135789960529, −10.728400591491459718076553949440, −10.20003443755974909118777989387, −9.08393213890814804134776580080, −8.74487528288751650697686471411, −7.63334046168389721590462250714, −7.00934500833344427274930621178, −6.759877204588653339641561320371, −5.42167489594091326272785427994, −4.79783000203054851331302000009, −3.83084386538503744221531940814, −3.36841797965808175888180829282, −2.58308717730526829579187680488, −1.27029878594246462346974539022, −0.32101028766438701531701004298, 0.14628795818129739149962978208, 1.59049956917480848297945843609, 2.318051905220684482887337840135, 3.237708740320030315815742833383, 4.0951289300528889004523436054, 4.62023414047068389680681019262, 5.56405893274105901996499638535, 6.42567742128295982691681075100, 7.076098228901495809045921430322, 7.83034992970720397962126285111, 8.60109240735776719229346198906, 9.19237379077202085840602740963, 9.9837660225873770355693183311, 10.6981520768529739320422308737, 11.65841483076456314990756095326, 12.17823781528114130022649163814, 12.65745867455356006007552800412, 13.26612112109137370676088684190, 14.630799077380810903840682345686, 15.03361343412907682109022980183, 15.27581270058586597507127225453, 16.378818983269832975202182646273, 16.86355833387798574509044671605, 17.48747692726183358364877405762, 18.523044718869038752600123992382

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