↑

Properties

Label 1-6017-6017.4619-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.859 + 0.511i$
Analytic cond. $27.9428$
Root an. cond. $27.9428$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.859 + 0.511i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (4619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.859 + 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3223712247 + 1.170607384i\)
\(L(\frac12)\) \(\approx\) \(0.3223712247 + 1.170607384i\)
\(L(1)\) \(\approx\) \(0.5729970264 + 0.4874307825i\)
\(L(1)\) \(\approx\) \(0.5729970264 + 0.4874307825i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.900 + 0.433i)T \)
3 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (0.222 + 0.974i)T \)
7 \( 1 + (0.623 + 0.781i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
17 \( 1 + (-0.222 - 0.974i)T \)
19 \( 1 + (0.900 + 0.433i)T \)
23 \( 1 + (0.900 + 0.433i)T \)
29 \( 1 + (0.222 - 0.974i)T \)
31 \( 1 + (-0.623 + 0.781i)T \)
37 \( 1 + (0.222 + 0.974i)T \)
41 \( 1 + T \)
43 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.900 - 0.433i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.222 - 0.974i)T \)
67 \( 1 + (-0.222 - 0.974i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.900 - 0.433i)T \)
79 \( 1 + (0.623 + 0.781i)T \)
83 \( 1 + T \)
89 \( 1 + (0.222 - 0.974i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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