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Properties

Label 1-4008-4008.869-r0-0-0
Degree $1$
Conductor $4008$
Sign $0.632 + 0.774i$
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  + (−0.929 − 0.369i)5-s + (0.776 + 0.629i)7-s + (−0.387 − 0.922i)11-s + (0.280 + 0.959i)13-s + (0.351 + 0.936i)17-s + (0.999 − 0.0378i)19-s + (0.974 − 0.225i)23-s + (0.726 + 0.686i)25-s + (0.974 + 0.225i)29-s + (−0.881 + 0.472i)31-s + (−0.489 − 0.872i)35-s + (−0.455 − 0.890i)37-s + (0.553 − 0.832i)41-s + (0.0567 + 0.998i)43-s + (−0.954 + 0.298i)47-s + ⋯
L(s)  = 1  + (−0.929 − 0.369i)5-s + (0.776 + 0.629i)7-s + (−0.387 − 0.922i)11-s + (0.280 + 0.959i)13-s + (0.351 + 0.936i)17-s + (0.999 − 0.0378i)19-s + (0.974 − 0.225i)23-s + (0.726 + 0.686i)25-s + (0.974 + 0.225i)29-s + (−0.881 + 0.472i)31-s + (−0.489 − 0.872i)35-s + (−0.455 − 0.890i)37-s + (0.553 − 0.832i)41-s + (0.0567 + 0.998i)43-s + (−0.954 + 0.298i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.632 + 0.774i$
Analytic conductor: \(18.6130\)
Root analytic conductor: \(18.6130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (869, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (0:\ ),\ 0.632 + 0.774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.428428944 + 0.6773506758i\)
\(L(\frac12)\) \(\approx\) \(1.428428944 + 0.6773506758i\)
\(L(1)\) \(\approx\) \(1.040609046 + 0.1074288328i\)
\(L(1)\) \(\approx\) \(1.040609046 + 0.1074288328i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.929 - 0.369i)T \)
7 \( 1 + (0.776 + 0.629i)T \)
11 \( 1 + (-0.387 - 0.922i)T \)
13 \( 1 + (0.280 + 0.959i)T \)
17 \( 1 + (0.351 + 0.936i)T \)
19 \( 1 + (0.999 - 0.0378i)T \)
23 \( 1 + (0.974 - 0.225i)T \)
29 \( 1 + (0.974 + 0.225i)T \)
31 \( 1 + (-0.881 + 0.472i)T \)
37 \( 1 + (-0.455 - 0.890i)T \)
41 \( 1 + (0.553 - 0.832i)T \)
43 \( 1 + (0.0567 + 0.998i)T \)
47 \( 1 + (-0.954 + 0.298i)T \)
53 \( 1 + (0.914 + 0.404i)T \)
59 \( 1 + (-0.351 + 0.936i)T \)
61 \( 1 + (-0.988 + 0.150i)T \)
67 \( 1 + (0.929 - 0.369i)T \)
71 \( 1 + (-0.243 - 0.969i)T \)
73 \( 1 + (0.644 - 0.764i)T \)
79 \( 1 + (-0.862 + 0.505i)T \)
83 \( 1 + (0.843 + 0.537i)T \)
89 \( 1 + (0.169 - 0.985i)T \)
97 \( 1 + (-0.881 - 0.472i)T \)
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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

−18.29768428356862014426592680184, −17.83323694057612876514821789764, −17.10290987579122524082963053032, −16.224682743055959657212860994786, −15.60767014729013166929856770220, −15.01739501704558050248902661029, −14.42289427744534287594847544048, −13.63203493877093364112484878394, −12.92297706777447385705943672726, −12.06083096957256664562931927920, −11.516724959961187202731820903798, −10.851965742524573942565182982924, −10.1929833792429038519711387476, −9.49308247810643600336810133574, −8.360834118772094080825476399927, −7.844683959459998889494273163723, −7.26661343830333864074506133270, −6.73776569277859560610416103604, −5.32645106184349186216041537244, −4.9747552108870029254976302990, −4.090872313968273246401893763002, −3.27698320707565633306350990509, −2.63429978723375214560922003056, −1.3967161324392709512101277433, −0.57120031400265459277115650987, 0.939894345945397091837136580657, 1.651972797688920599852095457972, 2.81921403489719734662860861721, 3.524275022438221632566051057634, 4.352805439248743754804641394097, 5.10816224862260054612686947817, 5.71431791043333869894586763278, 6.67260605929053496732257486571, 7.57890481967951936007522305212, 8.11786088203503548150039421405, 8.8887363954923908221506890444, 9.18770743609691764251869431624, 10.64156531769768503280295790545, 10.9952525081265418942851864865, 11.75866909083729005744973699213, 12.272499464527653388571900621416, 12.99920533277271497360940919950, 13.94705288277627288910725044801, 14.49477217804687655849730724345, 15.225542160505648423477001434818, 15.952073187765375059911402514780, 16.39453534166974660669889775314, 17.104369170718395835627003999304, 18.11797401558910180045956583393, 18.51015332512969945540994042942

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