Properties

Degree $1$
Conductor $4008$
Sign $0.996 - 0.0862i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.421 − 0.906i)5-s + (0.455 − 0.890i)7-s + (−0.929 − 0.369i)11-s + (−0.752 − 0.658i)13-s + (−0.881 − 0.472i)17-s + (−0.993 + 0.113i)19-s + (−0.776 + 0.629i)23-s + (−0.644 + 0.764i)25-s + (0.776 + 0.629i)29-s + (0.0944 − 0.995i)31-s + (−0.999 − 0.0378i)35-s + (0.988 + 0.150i)37-s + (−0.982 − 0.188i)41-s + (0.169 + 0.985i)43-s + (0.614 − 0.788i)47-s + ⋯
L(s,χ)  = 1  + (−0.421 − 0.906i)5-s + (0.455 − 0.890i)7-s + (−0.929 − 0.369i)11-s + (−0.752 − 0.658i)13-s + (−0.881 − 0.472i)17-s + (−0.993 + 0.113i)19-s + (−0.776 + 0.629i)23-s + (−0.644 + 0.764i)25-s + (0.776 + 0.629i)29-s + (0.0944 − 0.995i)31-s + (−0.999 − 0.0378i)35-s + (0.988 + 0.150i)37-s + (−0.982 − 0.188i)41-s + (0.169 + 0.985i)43-s + (0.614 − 0.788i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.996 - 0.0862i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.996 - 0.0862i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.996 - 0.0862i$
Motivic weight: \(0\)
Character: $\chi_{4008} (59, \cdot )$
Sato-Tate group: $\mu(166)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (1:\ ),\ 0.996 - 0.0862i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.4170065393 + 0.01802003099i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.4170065393 + 0.01802003099i\)
\(L(\chi,1)\) \(\approx\) \(0.6776911060 - 0.2602695099i\)
\(L(1,\chi)\) \(\approx\) \(0.6776911060 - 0.2602695099i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.34797416372347653959477340814, −17.769397962708672453004210778501, −17.12269861208392454304595054637, −16.06444880490223730450648762285, −15.45834468133115452903628042491, −15.01952755646954104797949513559, −14.367175657055849053890689412164, −13.675672065843983270566050747042, −12.62432803665996934211543245980, −12.18398991161242424046014782676, −11.430808307110770130254174134565, −10.71245148369686380337577249549, −10.20115661021908855307587187186, −9.292387659005131780575171232265, −8.368536826555618704624732070642, −7.99928279493949384324981211097, −6.98815289426292123164685885753, −6.46750086671497908679486740543, −5.64996676081205783329872699809, −4.59994334182707521684430575613, −4.26089052132411989480271571468, −2.96043076589750443782008037469, −2.363126833185861403426248862255, −1.85620490963919527203017874595, −0.128580971824125720165555227500, 0.39277037890671800599829084349, 1.33425350203524442904475025138, 2.31995043936633161120723009336, 3.20320841149273514968651830435, 4.334348014194577991464678050064, 4.5739745202820132158178782293, 5.444892996123636545516108718195, 6.24787138268022889172602804238, 7.44162529828319033299663429747, 7.719944277827127307202638359077, 8.47191662707359221465374712874, 9.17174713652738170600705105832, 10.19589253205555586029646664359, 10.59272721257057640824516245757, 11.51016563630464801772496972322, 12.07001041248912909618072147652, 13.07678833176771361570956489884, 13.28728955454593337283158721696, 14.11204034709411792634252759704, 15.117129052544211848226006443723, 15.49845536841293140133628685125, 16.44486220033695790772213349305, 16.80728236326312139229652117633, 17.64210360875275132003953230879, 18.106106013321252933992384975957

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