Properties

Degree $1$
Conductor $6017$
Sign $0.906 - 0.422i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.485 + 0.873i)2-s + (0.309 + 0.951i)3-s + (−0.527 + 0.849i)4-s + (−0.262 + 0.964i)5-s + (−0.681 + 0.732i)6-s + (0.981 + 0.192i)7-s + (−0.998 − 0.0483i)8-s + (−0.809 + 0.587i)9-s + (−0.970 + 0.239i)10-s + (−0.970 − 0.239i)12-s + (−0.809 + 0.587i)13-s + (0.309 + 0.951i)14-s + (−0.998 + 0.0483i)15-s + (−0.443 − 0.896i)16-s + (0.0241 − 0.999i)17-s + (−0.906 − 0.421i)18-s + ⋯
L(s,χ)  = 1  + (0.485 + 0.873i)2-s + (0.309 + 0.951i)3-s + (−0.527 + 0.849i)4-s + (−0.262 + 0.964i)5-s + (−0.681 + 0.732i)6-s + (0.981 + 0.192i)7-s + (−0.998 − 0.0483i)8-s + (−0.809 + 0.587i)9-s + (−0.970 + 0.239i)10-s + (−0.970 − 0.239i)12-s + (−0.809 + 0.587i)13-s + (0.309 + 0.951i)14-s + (−0.998 + 0.0483i)15-s + (−0.443 − 0.896i)16-s + (0.0241 − 0.999i)17-s + (−0.906 − 0.421i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.906 - 0.422i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.906 - 0.422i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.906 - 0.422i$
Motivic weight: \(0\)
Character: $\chi_{6017} (3657, \cdot )$
Sato-Tate group: $\mu(65)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.906 - 0.422i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(-0.8660172263 + 0.1920366313i\)
\(L(\frac12,\chi)\) \(\approx\) \(-0.8660172263 + 0.1920366313i\)
\(L(\chi,1)\) \(\approx\) \(0.4210367956 + 1.047516315i\)
\(L(1,\chi)\) \(\approx\) \(0.4210367956 + 1.047516315i\)

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

−17.42084248821005659141827276886, −16.63397768435814966845685059898, −15.50935615876999876778609749443, −14.85595036760485264483279160327, −14.39307895150077251836966258391, −13.56228135376981597930208812271, −13.14646116610472604873083399568, −12.349435293333240642337133776722, −12.11206155902585150703830592012, −11.34779480721274722624698886894, −10.694914984077240243606155867127, −9.80738229238869664126346327226, −8.97673044414064001420767423841, −8.50698287970627633272148291821, −7.73935540352013175030912718038, −7.19337803070201779921809494428, −5.932377619108045940727552928949, −5.48187321100284575562644159196, −4.74198586588287199645124615697, −3.99955104298967130820213517520, −3.277915566028028616630244040787, −2.1548373926638024762831608231, −1.848679578196444569487237431178, −0.950613067286231554517423310066, −0.18546054373875699046540690454, 1.8528198484021660822773905688, 2.65779210001324345375391192351, 3.46222371014093721700423995494, 3.95137276697316066344733183140, 4.8431633934089698463039089348, 5.30490998558048279685309082186, 5.98654084253930265346353164448, 7.15694919020060064826992850920, 7.43346304520207297262745616251, 8.16544868087498643404149460677, 8.92818684127602765081967892126, 9.62227963019587597179226699785, 10.23617847339512188165779233292, 11.246576455108075518796089671437, 11.68952072807965823256565871451, 12.22342471744085818004045273328, 13.61649344963205616894726720855, 13.97663047742809099063841557383, 14.51184818175834732239982770388, 15.07278800856705080113794015654, 15.42440940045596833500616120469, 16.46109096981025312196072720713, 16.57933622403593141371583786980, 17.633305197527342496028884476709, 18.1938992968114971196280044615

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