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Properties

Label 1-4008-4008.149-r0-0-0
Degree $1$
Conductor $4008$
Sign $0.713 - 0.700i$
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  + (0.700 − 0.713i)5-s + (0.898 + 0.438i)7-s + (−0.965 + 0.261i)11-s + (−0.982 + 0.188i)13-s + (0.280 − 0.959i)17-s + (0.521 + 0.853i)19-s + (0.988 − 0.150i)23-s + (−0.0189 − 0.999i)25-s + (0.988 + 0.150i)29-s + (−0.752 − 0.658i)31-s + (0.942 − 0.334i)35-s + (0.206 − 0.978i)37-s + (0.132 + 0.991i)41-s + (−0.999 + 0.0378i)43-s + (0.316 − 0.948i)47-s + ⋯
L(s)  = 1  + (0.700 − 0.713i)5-s + (0.898 + 0.438i)7-s + (−0.965 + 0.261i)11-s + (−0.982 + 0.188i)13-s + (0.280 − 0.959i)17-s + (0.521 + 0.853i)19-s + (0.988 − 0.150i)23-s + (−0.0189 − 0.999i)25-s + (0.988 + 0.150i)29-s + (−0.752 − 0.658i)31-s + (0.942 − 0.334i)35-s + (0.206 − 0.978i)37-s + (0.132 + 0.991i)41-s + (−0.999 + 0.0378i)43-s + (0.316 − 0.948i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.713 - 0.700i$
Analytic conductor: \(18.6130\)
Root analytic conductor: \(18.6130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (0:\ ),\ 0.713 - 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.910550020 - 0.7809548084i\)
\(L(\frac12)\) \(\approx\) \(1.910550020 - 0.7809548084i\)
\(L(1)\) \(\approx\) \(1.255777341 - 0.1663417214i\)
\(L(1)\) \(\approx\) \(1.255777341 - 0.1663417214i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.700 - 0.713i)T \)
7 \( 1 + (0.898 + 0.438i)T \)
11 \( 1 + (-0.965 + 0.261i)T \)
13 \( 1 + (-0.982 + 0.188i)T \)
17 \( 1 + (0.280 - 0.959i)T \)
19 \( 1 + (0.521 + 0.853i)T \)
23 \( 1 + (0.988 - 0.150i)T \)
29 \( 1 + (0.988 + 0.150i)T \)
31 \( 1 + (-0.752 - 0.658i)T \)
37 \( 1 + (0.206 - 0.978i)T \)
41 \( 1 + (0.132 + 0.991i)T \)
43 \( 1 + (-0.999 + 0.0378i)T \)
47 \( 1 + (0.316 - 0.948i)T \)
53 \( 1 + (0.243 + 0.969i)T \)
59 \( 1 + (-0.280 - 0.959i)T \)
61 \( 1 + (0.584 + 0.811i)T \)
67 \( 1 + (-0.700 - 0.713i)T \)
71 \( 1 + (0.351 - 0.936i)T \)
73 \( 1 + (-0.0567 - 0.998i)T \)
79 \( 1 + (0.169 - 0.985i)T \)
83 \( 1 + (-0.929 - 0.369i)T \)
89 \( 1 + (0.993 + 0.113i)T \)
97 \( 1 + (-0.752 + 0.658i)T \)
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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

−18.46681364333020375154258688502, −17.77631087123141032336933822610, −17.35497553516858124926391758902, −16.76341467952156255508071281481, −15.68254205816721391216207475538, −15.06500891072729477575062115460, −14.47464542490609029443936607174, −13.85553035816803003691517962641, −13.19200182489896767771911236906, −12.516793548542669546189654591739, −11.48025314168082186430565197909, −10.938699211233102012410456674955, −10.27959514324405975930197117524, −9.819658260590610553371987706081, −8.74449645381778900381921550878, −8.08390711051461974935775331535, −7.21617138246905405031446449865, −6.8587165272074242758863615096, −5.657193939565581665628959602648, −5.20452063379981406742804894659, −4.46776702972986106130552857881, −3.276220151945300466612038748468, −2.69643990762205146885934293901, −1.87101951861211153127659047010, −0.94286661617944829870565661495, 0.6538987233971629616120019617, 1.695308998947403071531771688637, 2.33522526477888544297032892609, 3.09075851638542903059243624303, 4.45110756036078646192884590562, 5.06798244068103140509848636307, 5.35724865173791628961830316133, 6.30426814150593125446209305348, 7.45793036430728444848021169970, 7.7985084968850438043038716869, 8.76792932984786178916298098337, 9.343576168528341557194948550758, 10.05042678321783469109079557075, 10.71340202527140573732306006081, 11.73957891734185399195455527386, 12.17062833826039041052557376204, 12.92692364809960300511891207368, 13.601501223260473244353285309758, 14.37007160563986707805913910164, 14.86549063818371530858382958306, 15.7222104595336575411592796872, 16.5212578656808136417136671065, 16.944331228698942338327344381987, 17.912918904448930266877286678612, 18.17630899062664951630967781157

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