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Properties

Label 1-4008-4008.197-r0-0-0
Degree $1$
Conductor $4008$
Sign $0.0884 - 0.996i$
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  + (0.387 − 0.922i)5-s + (0.974 − 0.225i)7-s + (0.132 − 0.991i)11-s + (−0.0944 + 0.995i)13-s + (−0.800 − 0.599i)17-s + (−0.489 − 0.872i)19-s + (0.997 + 0.0756i)23-s + (−0.700 − 0.713i)25-s + (0.997 − 0.0756i)29-s + (0.351 + 0.936i)31-s + (0.169 − 0.985i)35-s + (0.776 + 0.629i)37-s + (−0.752 + 0.658i)41-s + (−0.0189 + 0.999i)43-s + (0.584 − 0.811i)47-s + ⋯
L(s)  = 1  + (0.387 − 0.922i)5-s + (0.974 − 0.225i)7-s + (0.132 − 0.991i)11-s + (−0.0944 + 0.995i)13-s + (−0.800 − 0.599i)17-s + (−0.489 − 0.872i)19-s + (0.997 + 0.0756i)23-s + (−0.700 − 0.713i)25-s + (0.997 − 0.0756i)29-s + (0.351 + 0.936i)31-s + (0.169 − 0.985i)35-s + (0.776 + 0.629i)37-s + (−0.752 + 0.658i)41-s + (−0.0189 + 0.999i)43-s + (0.584 − 0.811i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0884 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0884 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.0884 - 0.996i$
Analytic conductor: \(18.6130\)
Root analytic conductor: \(18.6130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (0:\ ),\ 0.0884 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.583223009 - 1.448816254i\)
\(L(\frac12)\) \(\approx\) \(1.583223009 - 1.448816254i\)
\(L(1)\) \(\approx\) \(1.234661726 - 0.3882711698i\)
\(L(1)\) \(\approx\) \(1.234661726 - 0.3882711698i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.387 - 0.922i)T \)
7 \( 1 + (0.974 - 0.225i)T \)
11 \( 1 + (0.132 - 0.991i)T \)
13 \( 1 + (-0.0944 + 0.995i)T \)
17 \( 1 + (-0.800 - 0.599i)T \)
19 \( 1 + (-0.489 - 0.872i)T \)
23 \( 1 + (0.997 + 0.0756i)T \)
29 \( 1 + (0.997 - 0.0756i)T \)
31 \( 1 + (0.351 + 0.936i)T \)
37 \( 1 + (0.776 + 0.629i)T \)
41 \( 1 + (-0.752 + 0.658i)T \)
43 \( 1 + (-0.0189 + 0.999i)T \)
47 \( 1 + (0.584 - 0.811i)T \)
53 \( 1 + (-0.614 - 0.788i)T \)
59 \( 1 + (0.800 - 0.599i)T \)
61 \( 1 + (0.455 + 0.890i)T \)
67 \( 1 + (-0.387 - 0.922i)T \)
71 \( 1 + (0.822 + 0.569i)T \)
73 \( 1 + (-0.726 + 0.686i)T \)
79 \( 1 + (0.644 - 0.764i)T \)
83 \( 1 + (0.982 - 0.188i)T \)
89 \( 1 + (-0.0567 - 0.998i)T \)
97 \( 1 + (0.351 - 0.936i)T \)
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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

−18.61115275709503437499148398606, −17.71953632306355758045797740711, −17.56702508490909209239541025701, −16.886703965351647698226484362224, −15.5784116068008230966725938428, −15.12804688158151881117939829498, −14.716934008417420186765686166058, −13.99706831428144935569793149078, −13.18405079331871885656167534837, −12.45836772143318285809916686579, −11.78202405658240358072396641789, −10.754893333204359922452924656801, −10.628417397164982124953025641290, −9.74845985323615052094529767910, −8.907737764807965961975157681606, −8.06987256215018325766976502932, −7.497690387235421430493626440752, −6.69208288952451084635743191292, −5.978656803814803428543354968841, −5.2111185254917122407066196749, −4.4073249631544750861324374290, −3.625982819833176142942436057241, −2.47393120463044916347605365687, −2.14855634230838368442221683888, −1.097474392184406673671387878911, 0.65769506257119343492039331645, 1.387898023351859140925437816619, 2.24728208976655508340724854934, 3.140197572573241070621806900432, 4.42478441470879297323813040048, 4.675976381063197576493523339612, 5.39430797749904124749002374175, 6.46799867279037504605627826394, 6.929868028874300169182450680970, 8.11040833307416465866330738115, 8.62910190503532187248781884789, 9.093222073829739084969649469754, 9.945348859948593369229251904861, 10.92438658566554144908223675198, 11.4609365881218614781648778473, 11.97491281158903426929156398907, 13.079483271646709896778770899929, 13.516082217912449017618764295016, 14.10842167552755750641474968268, 14.83892207675985349652910765306, 15.73657243439954180090940740262, 16.38689983192894498251092877860, 16.97009085755119329907393238576, 17.57986045119076965723663531287, 18.1550764153446214667485994856

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