Properties

Degree 1
Conductor $ 11 \cdot 547 $
Sign $-0.859 - 0.511i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.900 − 0.433i)2-s + (−0.623 − 0.781i)3-s + (0.623 + 0.781i)4-s + (0.222 − 0.974i)5-s + (0.222 + 0.974i)6-s + (0.623 − 0.781i)7-s + (−0.222 − 0.974i)8-s + (−0.222 + 0.974i)9-s + (−0.623 + 0.781i)10-s + (0.222 − 0.974i)12-s + (0.900 − 0.433i)13-s + (−0.900 + 0.433i)14-s + (−0.900 + 0.433i)15-s + (−0.222 + 0.974i)16-s + (−0.222 + 0.974i)17-s + (0.623 − 0.781i)18-s + ⋯
L(s,χ)  = 1  + (−0.900 − 0.433i)2-s + (−0.623 − 0.781i)3-s + (0.623 + 0.781i)4-s + (0.222 − 0.974i)5-s + (0.222 + 0.974i)6-s + (0.623 − 0.781i)7-s + (−0.222 − 0.974i)8-s + (−0.222 + 0.974i)9-s + (−0.623 + 0.781i)10-s + (0.222 − 0.974i)12-s + (0.900 − 0.433i)13-s + (−0.900 + 0.433i)14-s + (−0.900 + 0.433i)15-s + (−0.222 + 0.974i)16-s + (−0.222 + 0.974i)17-s + (0.623 − 0.781i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.859 - 0.511i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.859 - 0.511i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6017\)    =    \(11 \cdot 547\)
\( \varepsilon \)  =  $-0.859 - 0.511i$
motivic weight  =  \(0\)
character  :  $\chi_{6017} (538, \cdot )$
Sato-Tate  :  $\mu(14)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 6017,\ (0:\ ),\ -0.859 - 0.511i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.3223712247 - 1.170607384i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.3223712247 - 1.170607384i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5729970264 - 0.4874307825i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5729970264 - 0.4874307825i\)

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

−17.89203973683156865087978049195, −17.51078098466627117632239199117, −16.66231739075129132479277318179, −15.974941512547660907459658939521, −15.48072499936310831826753512748, −15.05497202743504506324642458395, −14.144091632935084501224906061237, −13.89308671792419237276836308319, −12.40103470510844524045940642383, −11.607426542992125295589645325396, −11.118369811147792811009839821198, −10.89844489181034545509171446368, −9.81348147770680446568984374178, −9.44322116414122741137669766503, −8.849373358153230320338434054065, −7.92225220468446486218718260193, −7.24281659073960440505512138836, −6.39492296963645154181166021924, −5.97238537932928837258399938702, −5.25791881241809373097658777014, −4.59327866058119118651528545947, −3.37254421372031213617078825710, −2.76265087844673982194621297354, −1.77701125777360349572690171497, −0.905182875491812074391899891445, 0.64095834983040384530251084088, 1.06554374463011581851791313922, 1.70040151978337177833678178603, 2.53304324980962025602090183138, 3.67798173469025564239267768969, 4.40370756543719982124348882664, 5.32052767764538486651813263103, 5.99785785573516968342178281090, 6.81006922598783121760070457219, 7.66009020723303161268999665967, 7.90361869736717696614744478365, 8.86956539494496218929291888482, 9.20283674991687006812379541802, 10.480266959577197240479566972199, 10.78959569447400446613549278264, 11.310349645890915964668928719851, 12.23621680184761998943927680966, 12.66015213674914924194206531790, 13.31500692148016080431098993442, 13.799703525628851299045660028813, 14.89492399738562931998050174215, 15.96702645943713696838542950119, 16.31302905265971235020957767897, 17.17798725985599368864388452115, 17.33251081854049703623314116839

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