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Properties

Label 1-4008-4008.941-r0-0-0
Degree $1$
Conductor $4008$
Sign $0.504 - 0.863i$
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  + (0.800 − 0.599i)5-s + (0.862 − 0.505i)7-s + (0.672 + 0.739i)11-s + (0.954 − 0.298i)13-s + (0.898 + 0.438i)17-s + (−0.929 − 0.369i)19-s + (−0.644 + 0.764i)23-s + (0.280 − 0.959i)25-s + (−0.644 − 0.764i)29-s + (0.206 − 0.978i)31-s + (0.387 − 0.922i)35-s + (−0.0189 + 0.999i)37-s + (−0.914 − 0.404i)41-s + (−0.843 − 0.537i)43-s + (0.993 − 0.113i)47-s + ⋯
L(s)  = 1  + (0.800 − 0.599i)5-s + (0.862 − 0.505i)7-s + (0.672 + 0.739i)11-s + (0.954 − 0.298i)13-s + (0.898 + 0.438i)17-s + (−0.929 − 0.369i)19-s + (−0.644 + 0.764i)23-s + (0.280 − 0.959i)25-s + (−0.644 − 0.764i)29-s + (0.206 − 0.978i)31-s + (0.387 − 0.922i)35-s + (−0.0189 + 0.999i)37-s + (−0.914 − 0.404i)41-s + (−0.843 − 0.537i)43-s + (0.993 − 0.113i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.504 - 0.863i$
Analytic conductor: \(18.6130\)
Root analytic conductor: \(18.6130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (941, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (0:\ ),\ 0.504 - 0.863i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.236051541 - 1.283706991i\)
\(L(\frac12)\) \(\approx\) \(2.236051541 - 1.283706991i\)
\(L(1)\) \(\approx\) \(1.435004217 - 0.3070074442i\)
\(L(1)\) \(\approx\) \(1.435004217 - 0.3070074442i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.800 - 0.599i)T \)
7 \( 1 + (0.862 - 0.505i)T \)
11 \( 1 + (0.672 + 0.739i)T \)
13 \( 1 + (0.954 - 0.298i)T \)
17 \( 1 + (0.898 + 0.438i)T \)
19 \( 1 + (-0.929 - 0.369i)T \)
23 \( 1 + (-0.644 + 0.764i)T \)
29 \( 1 + (-0.644 - 0.764i)T \)
31 \( 1 + (0.206 - 0.978i)T \)
37 \( 1 + (-0.0189 + 0.999i)T \)
41 \( 1 + (-0.914 - 0.404i)T \)
43 \( 1 + (-0.843 - 0.537i)T \)
47 \( 1 + (0.993 - 0.113i)T \)
53 \( 1 + (0.521 - 0.853i)T \)
59 \( 1 + (-0.898 + 0.438i)T \)
61 \( 1 + (-0.0567 - 0.998i)T \)
67 \( 1 + (-0.800 - 0.599i)T \)
71 \( 1 + (0.776 - 0.629i)T \)
73 \( 1 + (0.752 - 0.658i)T \)
79 \( 1 + (-0.553 + 0.832i)T \)
83 \( 1 + (-0.822 - 0.569i)T \)
89 \( 1 + (-0.132 - 0.991i)T \)
97 \( 1 + (0.206 + 0.978i)T \)
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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

−18.41632046626682454840114480206, −18.20039648815450517720618360788, −17.174619252949726691081296805319, −16.686851391773816166479691041304, −15.935979915964329546287651723549, −14.98694562320259681387726595565, −14.32296895423103352035654286248, −14.10916024789370190660364353075, −13.255556272274748614702591877757, −12.32846327120743029918100960326, −11.69985396339823066381827781439, −10.88258539275398164833851307735, −10.5253846537958661938405244383, −9.5403222453870128813344453259, −8.75436348477821707610068935440, −8.37310924951106611543527811936, −7.32470257231426303344227366618, −6.50378127345251387927354468728, −5.89691394897678097889539059943, −5.34715946249292196705207649617, −4.29679436833591908875257756902, −3.48110649955697241113763565478, −2.67214320235254882135966596075, −1.746564384627749912475449084621, −1.19632471830831560772470808512, 0.7395934857090618047331952663, 1.73316739949137760703887519839, 1.958793178603529515957491110031, 3.44378275443961236680421877355, 4.16095290759106883398819914380, 4.8257071905158938682310564053, 5.69324265207207492290082639845, 6.24943440534231340606837257664, 7.16792506822201428660140867151, 8.06461882856784535240138325416, 8.51674995378306535441893767936, 9.428955582212865930006884552327, 10.05215852917526059908610848239, 10.67907326531082246211635026007, 11.58187006255864200393434409472, 12.14798722091798923696581979493, 13.04312479569781131839981733369, 13.612044181724976725636998052660, 14.12451301663345193820201367056, 15.08576997470327405583331559003, 15.41434688360311093111648344202, 16.782815572101356283165593669132, 16.94084392007729416047259906732, 17.533495394821826555603020955023, 18.27201211742160594067216381758

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