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Properties

Label 1-6017-6017.332-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.985 + 0.170i$
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  + (0.348 + 0.937i)2-s + (0.691 − 0.722i)3-s + (−0.757 + 0.652i)4-s + (0.729 + 0.683i)5-s + (0.918 + 0.396i)6-s + (−0.430 + 0.902i)7-s + (−0.875 − 0.482i)8-s + (−0.0448 − 0.998i)9-s + (−0.386 + 0.922i)10-s + (−0.0517 + 0.998i)12-s + (0.393 − 0.919i)13-s + (−0.995 − 0.0896i)14-s + (0.998 − 0.0552i)15-s + (0.147 − 0.989i)16-s + (0.999 + 0.0276i)17-s + (0.920 − 0.389i)18-s + ⋯
L(s)  = 1  + (0.348 + 0.937i)2-s + (0.691 − 0.722i)3-s + (−0.757 + 0.652i)4-s + (0.729 + 0.683i)5-s + (0.918 + 0.396i)6-s + (−0.430 + 0.902i)7-s + (−0.875 − 0.482i)8-s + (−0.0448 − 0.998i)9-s + (−0.386 + 0.922i)10-s + (−0.0517 + 0.998i)12-s + (0.393 − 0.919i)13-s + (−0.995 − 0.0896i)14-s + (0.998 − 0.0552i)15-s + (0.147 − 0.989i)16-s + (0.999 + 0.0276i)17-s + (0.920 − 0.389i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.985 + 0.170i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (332, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.985 + 0.170i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.654811413 + 0.2280640104i\)
\(L(\frac12)\) \(\approx\) \(2.654811413 + 0.2280640104i\)
\(L(1)\) \(\approx\) \(1.491865427 + 0.4751550280i\)
\(L(1)\) \(\approx\) \(1.491865427 + 0.4751550280i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.348 + 0.937i)T \)
3 \( 1 + (0.691 - 0.722i)T \)
5 \( 1 + (0.729 + 0.683i)T \)
7 \( 1 + (-0.430 + 0.902i)T \)
13 \( 1 + (0.393 - 0.919i)T \)
17 \( 1 + (0.999 + 0.0276i)T \)
19 \( 1 + (0.209 - 0.977i)T \)
23 \( 1 + (-0.675 + 0.736i)T \)
29 \( 1 + (-0.821 + 0.570i)T \)
31 \( 1 + (0.817 - 0.576i)T \)
37 \( 1 + (-0.373 - 0.927i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (0.962 + 0.272i)T \)
47 \( 1 + (-0.607 - 0.794i)T \)
53 \( 1 + (-0.775 + 0.631i)T \)
59 \( 1 + (0.906 - 0.421i)T \)
61 \( 1 + (0.590 - 0.806i)T \)
67 \( 1 + (0.0517 - 0.998i)T \)
71 \( 1 + (0.380 - 0.924i)T \)
73 \( 1 + (0.840 - 0.542i)T \)
79 \( 1 + (-0.618 - 0.786i)T \)
83 \( 1 + (-0.989 + 0.144i)T \)
89 \( 1 + (0.952 + 0.305i)T \)
97 \( 1 + (0.161 - 0.986i)T \)
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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

−17.64229376890035370239897032502, −16.98738083771413871927572581750, −16.28295100799502135631851561467, −15.93934513500359018583180063158, −14.52979381866704702703832922688, −14.40963916425452842872535731611, −13.70079900022542376181435206628, −13.242405925922344569042820203789, −12.52367205755749815092071608888, −11.785961167750775283193931015481, −10.94193306637907222486000047543, −10.18962522820531586958318843163, −9.78458406052277314240730793256, −9.449271394549499364676923980872, −8.41817132279106342853491321937, −8.07790168801221665932606186785, −6.78040156215115394056547937860, −5.907695402346691856609882795019, −5.29154420170516690565766181181, −4.26617985582528836749538828265, −4.15314084107985523712404955568, −3.24300166811592301324798097041, −2.50481335000458448402821821685, −1.60405418812668849554664746312, −1.04576108109108176313565795224, 0.55778811713768791379509782482, 1.83013424708283773317236390817, 2.6320811189881328563762350902, 3.266717915607198314274394540896, 3.74568249372184763013934952378, 5.208205242716217191622001686396, 5.69601128874776488722214865502, 6.23899551400347505997962055843, 6.9167410854964834450707303493, 7.64193446043175655970468471157, 8.119923580654316017857597084, 9.11852174766739243270880959871, 9.37068258729946030186218499702, 10.16973935040865538237062119850, 11.279347071198052127754122642, 12.14307407826675002238692702111, 12.75899792117171784084922793100, 13.23274106219720529892727278107, 13.93444085188787908982430566648, 14.37232937958638070710403240914, 15.16650688359628076720056575710, 15.486545523140402640706614157472, 16.23368872095346481295327764061, 17.33994243756834052854023443139, 17.70053551048407281849963447529

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