Properties

Degree 1
Conductor $ 2^{3} \cdot 3 \cdot 167 $
Sign $0.905 - 0.423i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.999 + 0.0378i)5-s + (−0.243 − 0.969i)7-s + (0.489 + 0.872i)11-s + (0.726 + 0.686i)13-s + (0.421 − 0.906i)17-s + (0.584 − 0.811i)19-s + (0.822 + 0.569i)23-s + (0.997 + 0.0756i)25-s + (0.822 − 0.569i)29-s + (−0.965 − 0.261i)31-s + (−0.206 − 0.978i)35-s + (0.672 − 0.739i)37-s + (0.862 + 0.505i)41-s + (0.988 + 0.150i)43-s + (−0.280 + 0.959i)47-s + ⋯
L(s,χ)  = 1  + (0.999 + 0.0378i)5-s + (−0.243 − 0.969i)7-s + (0.489 + 0.872i)11-s + (0.726 + 0.686i)13-s + (0.421 − 0.906i)17-s + (0.584 − 0.811i)19-s + (0.822 + 0.569i)23-s + (0.997 + 0.0756i)25-s + (0.822 − 0.569i)29-s + (−0.965 − 0.261i)31-s + (−0.206 − 0.978i)35-s + (0.672 − 0.739i)37-s + (0.862 + 0.505i)41-s + (0.988 + 0.150i)43-s + (−0.280 + 0.959i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.905 - 0.423i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.905 - 0.423i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $0.905 - 0.423i$
motivic weight  =  \(0\)
character  :  $\chi_{4008} (5, \cdot )$
Sato-Tate  :  $\mu(166)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4008,\ (0:\ ),\ 0.905 - 0.423i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(2.516243382 - 0.5596931224i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(2.516243382 - 0.5596931224i\)
\(L(\chi,1)\)  \(\approx\)  \(1.437778951 - 0.1323782767i\)
\(L(1,\chi)\)  \(\approx\)  \(1.437778951 - 0.1323782767i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.547028240371887934515456855783, −17.91783105997089792355068835064, −17.19412279077592235185997503018, −16.4231010114673191861810896799, −16.0012629839334261713127372433, −14.95874566063322869018644579241, −14.48905021515859735035928364364, −13.77029682734569786490375896417, −12.955614964928426932123010987881, −12.54438891408060651506453282971, −11.706875716884002648987829916937, −10.79443700407109937948985827508, −10.31056042125694592817987960784, −9.44420463933935306665732289404, −8.712238523219425627185690046304, −8.41962855699806421839839716656, −7.25071755245782460384959061438, −6.26866303570704001042891237488, −5.797738276771033415710801973212, −5.437416764699204209866513694304, −4.24163928748052474042002962447, −3.1871137485327577845751906413, −2.807636599750999843857403838670, −1.619271268564072795625343993045, −1.03919794462569215478759031452, 0.877250321573456491910240484348, 1.49815620188104816649697464542, 2.50173042716856027984566859387, 3.295514903703905968152155005, 4.27088626838140918294834750181, 4.85080065800751754821007535621, 5.7847070295079031840186702607, 6.574624350316206960966246076784, 7.11699571548580651360146591333, 7.75680006285122129898588332867, 9.03159367684395673583882027725, 9.5258138450941110770818622210, 9.86496507853284089564684897434, 11.079644189369532877677615922655, 11.223400116003614258368121874016, 12.44957268610107257066415660474, 13.0513602723646148398293892284, 13.761592983840892793854281512596, 14.16612017522865423081224831899, 14.85712098106693240905732342094, 15.957769746929328664726905324437, 16.36991703814022512556273399801, 17.20549461323274186596827525590, 17.70823529784630644861705762322, 18.211360342443510220129315682219

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