Properties

Degree $1$
Conductor $6017$
Sign $-0.658 + 0.753i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.503 − 0.863i)2-s + (−0.858 − 0.512i)3-s + (−0.492 + 0.870i)4-s + (−0.295 − 0.955i)5-s + (−0.0103 + 0.999i)6-s + (0.998 − 0.0552i)7-s + (0.999 − 0.0138i)8-s + (0.473 + 0.880i)9-s + (−0.675 + 0.736i)10-s + (0.868 − 0.495i)12-s + (0.963 − 0.266i)13-s + (−0.550 − 0.834i)14-s + (−0.235 + 0.971i)15-s + (−0.515 − 0.856i)16-s + (−0.618 + 0.786i)17-s + (0.521 − 0.853i)18-s + ⋯
L(s,χ)  = 1  + (−0.503 − 0.863i)2-s + (−0.858 − 0.512i)3-s + (−0.492 + 0.870i)4-s + (−0.295 − 0.955i)5-s + (−0.0103 + 0.999i)6-s + (0.998 − 0.0552i)7-s + (0.999 − 0.0138i)8-s + (0.473 + 0.880i)9-s + (−0.675 + 0.736i)10-s + (0.868 − 0.495i)12-s + (0.963 − 0.266i)13-s + (−0.550 − 0.834i)14-s + (−0.235 + 0.971i)15-s + (−0.515 − 0.856i)16-s + (−0.618 + 0.786i)17-s + (0.521 − 0.853i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.658 + 0.753i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.658 + 0.753i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.658 + 0.753i$
Motivic weight: \(0\)
Character: $\chi_{6017} (343, \cdot )$
Sato-Tate group: $\mu(910)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.658 + 0.753i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(-0.2329061980 - 0.5128235869i\)
\(L(\frac12,\chi)\) \(\approx\) \(-0.2329061980 - 0.5128235869i\)
\(L(\chi,1)\) \(\approx\) \(0.4455471530 - 0.4197721554i\)
\(L(1,\chi)\) \(\approx\) \(0.4455471530 - 0.4197721554i\)

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

−18.08517885367654774943423515946, −17.45835216392634159938454907038, −16.88404851946215383632969562000, −16.13523407559460150032300751973, −15.50189637029271261697094524034, −15.05069510003385143123968039535, −14.56976013559699969676583959863, −13.67875974666084085793171713849, −13.11424672190216040529079801138, −11.73185215451784312175805881966, −11.34552262125661222726067112285, −10.88368618608340690599607085771, −10.23883257545899983546713420592, −9.46959439630964823814248874096, −8.74818404250667568151363067241, −8.04345904953828835326466173168, −7.26367701607928230509283218703, −6.60674304151949710740176921614, −6.164415564031843783478725799319, −5.30285158295118401059248157069, −4.61697029225646302814150971701, −4.092666312217241956254503746985, −3.060381900191662771672935940615, −1.82388058961155239558229759532, −1.04027785219504768342637960876, 0.24308262887386059963002239846, 1.03457247577707936128560991809, 1.741584169311604809450811690, 2.25539538671753794969336785447, 3.6448465365072780290442684144, 4.413612668798809726125482571673, 4.733338334142689880307430631122, 5.70675223827815200376412747420, 6.452396854447269582031047508963, 7.46059105725979345229299256614, 8.18759218473501264885499211392, 8.4567104680545631730104381779, 9.19169295690753031000200270683, 10.37031076475100625791864823449, 10.74626568744805300832993879190, 11.35504192738203213473543962190, 11.93342000507249762540041092183, 12.58993864804887265902434410048, 13.10107726239834145977026725751, 13.58515658217034414820135201859, 14.60561591196965874481202821447, 15.59216438538690838679429306451, 16.240139937703633100814939300668, 16.98972390347184189003459616197, 17.35897779273341383134167692543

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