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Properties

Label 1-6017-6017.1047-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.456 - 0.889i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.853 + 0.520i)2-s + (−0.309 + 0.951i)3-s + (0.457 + 0.889i)4-s + (0.789 + 0.613i)5-s + (−0.759 + 0.650i)6-s + (−0.726 + 0.686i)7-s + (−0.0724 + 0.997i)8-s + (−0.809 − 0.587i)9-s + (0.354 + 0.935i)10-s + (−0.987 + 0.160i)12-s + (−0.913 + 0.406i)13-s + (−0.978 + 0.207i)14-s + (−0.827 + 0.561i)15-s + (−0.581 + 0.813i)16-s + (0.974 − 0.223i)17-s + (−0.384 − 0.923i)18-s + ⋯
L(s)  = 1  + (0.853 + 0.520i)2-s + (−0.309 + 0.951i)3-s + (0.457 + 0.889i)4-s + (0.789 + 0.613i)5-s + (−0.759 + 0.650i)6-s + (−0.726 + 0.686i)7-s + (−0.0724 + 0.997i)8-s + (−0.809 − 0.587i)9-s + (0.354 + 0.935i)10-s + (−0.987 + 0.160i)12-s + (−0.913 + 0.406i)13-s + (−0.978 + 0.207i)14-s + (−0.827 + 0.561i)15-s + (−0.581 + 0.813i)16-s + (0.974 − 0.223i)17-s + (−0.384 − 0.923i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.456 - 0.889i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1047, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.456 - 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.9907823330 + 0.6051537722i\)
\(L(\frac12)\) \(\approx\) \(-0.9907823330 + 0.6051537722i\)
\(L(1)\) \(\approx\) \(0.7002030495 + 1.107197676i\)
\(L(1)\) \(\approx\) \(0.7002030495 + 1.107197676i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.853 + 0.520i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.789 + 0.613i)T \)
7 \( 1 + (-0.726 + 0.686i)T \)
13 \( 1 + (-0.913 + 0.406i)T \)
17 \( 1 + (0.974 - 0.223i)T \)
19 \( 1 + (-0.594 + 0.804i)T \)
23 \( 1 + (0.0402 + 0.999i)T \)
29 \( 1 + (-0.715 - 0.698i)T \)
31 \( 1 + (-0.215 - 0.976i)T \)
37 \( 1 + (0.962 - 0.270i)T \)
41 \( 1 + (0.669 + 0.743i)T \)
43 \( 1 + (-0.632 - 0.774i)T \)
47 \( 1 + (-0.769 + 0.638i)T \)
53 \( 1 + (-0.984 + 0.176i)T \)
59 \( 1 + (-0.594 - 0.804i)T \)
61 \( 1 + (0.759 - 0.650i)T \)
67 \( 1 + (-0.632 - 0.774i)T \)
71 \( 1 + (-0.737 - 0.675i)T \)
73 \( 1 + (-0.999 + 0.0322i)T \)
79 \( 1 + (0.836 - 0.548i)T \)
83 \( 1 + (-0.136 + 0.990i)T \)
89 \( 1 + (-0.568 + 0.822i)T \)
97 \( 1 + (-0.999 + 0.0161i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.11751303207892542182433064670, −16.54691407049182851874930691593, −16.07675262656553016679741724781, −14.6932685310879951313036724247, −14.51294568556324890330538454483, −13.624902651280247209851233343535, −13.0354590141487572837397499328, −12.725992118292226157269760236159, −12.24682772969565845110384223915, −11.373274842331037376214111660648, −10.52048070262397340091646876646, −10.140024816354946158433339275680, −9.33345355978693020247668893648, −8.50122623516686628236215801939, −7.4428847374510789441814774664, −6.89836331170593772419458232809, −6.21875143299387515092243430404, −5.62311356044308403634664575879, −4.93863960626741894650535745044, −4.290787860079473281050080651016, −3.07705138657531310839767582904, −2.66939603307233789771003332802, −1.71482401688200311869818267497, −1.07256137290909166545553802775, −0.20716931903836148813149092440, 1.851453439135379830861201091277, 2.593298789349039561852134331844, 3.26691886061610375198987097942, 3.84370592565538747562647191441, 4.79391635525261240834480571006, 5.44442852358537586408041549551, 6.07537983622246953327314493105, 6.33385595169120178782277097075, 7.407358206356529708909705918880, 8.04255051196238973063511779587, 9.2824197685694039880240846218, 9.53792927168154967164784409427, 10.193505458912401066497096843102, 11.17829239757022639089309923648, 11.65025205856690354234910246560, 12.425619632176254261206398929104, 13.01034000874219192395996843244, 13.87170807649576373456754143532, 14.51284763633013484739573693328, 14.97038976290432513437379505115, 15.42734359695452391185681166967, 16.39460715886414616234653751142, 16.693594677092331768047596653268, 17.29988330285881381048977341609, 18.036512748807485454045474914802

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