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Properties

Label 1-2268-2268.2063-r1-0-0
Degree $1$
Conductor $2268$
Sign $-0.999 + 0.0297i$
Analytic cond. $243.730$
Root an. cond. $243.730$
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  + (0.973 + 0.230i)5-s + (−0.686 − 0.727i)11-s + (0.0581 − 0.998i)13-s + (0.173 − 0.984i)17-s + (0.173 + 0.984i)19-s + (0.597 + 0.802i)23-s + (0.893 + 0.448i)25-s + (−0.893 − 0.448i)29-s + (−0.993 + 0.116i)31-s + (0.173 − 0.984i)37-s + (−0.835 − 0.549i)41-s + (−0.973 + 0.230i)43-s + (0.993 + 0.116i)47-s + (0.5 + 0.866i)53-s + (−0.5 − 0.866i)55-s + ⋯
L(s)  = 1  + (0.973 + 0.230i)5-s + (−0.686 − 0.727i)11-s + (0.0581 − 0.998i)13-s + (0.173 − 0.984i)17-s + (0.173 + 0.984i)19-s + (0.597 + 0.802i)23-s + (0.893 + 0.448i)25-s + (−0.893 − 0.448i)29-s + (−0.993 + 0.116i)31-s + (0.173 − 0.984i)37-s + (−0.835 − 0.549i)41-s + (−0.973 + 0.230i)43-s + (0.993 + 0.116i)47-s + (0.5 + 0.866i)53-s + (−0.5 − 0.866i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.999 + 0.0297i$
Analytic conductor: \(243.730\)
Root analytic conductor: \(243.730\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (2063, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2268,\ (1:\ ),\ -0.999 + 0.0297i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005930854141 - 0.3981792728i\)
\(L(\frac12)\) \(\approx\) \(0.005930854141 - 0.3981792728i\)
\(L(1)\) \(\approx\) \(1.053605790 - 0.1010292595i\)
\(L(1)\) \(\approx\) \(1.053605790 - 0.1010292595i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.973 + 0.230i)T \)
11 \( 1 + (-0.686 - 0.727i)T \)
13 \( 1 + (0.0581 - 0.998i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.597 + 0.802i)T \)
29 \( 1 + (-0.893 - 0.448i)T \)
31 \( 1 + (-0.993 + 0.116i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (-0.835 - 0.549i)T \)
43 \( 1 + (-0.973 + 0.230i)T \)
47 \( 1 + (0.993 + 0.116i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.973 - 0.230i)T \)
61 \( 1 + (-0.597 + 0.802i)T \)
67 \( 1 + (0.0581 - 0.998i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (0.939 + 0.342i)T \)
79 \( 1 + (-0.893 - 0.448i)T \)
83 \( 1 + (0.835 - 0.549i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (0.686 + 0.727i)T \)
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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

−20.14941252007394755758108541748, −18.91474890615098118483842920338, −18.44045577005059196116226221091, −17.7037723835500652897906592835, −16.84938161279042449029161163257, −16.592629385650155255575291738739, −15.37958921217566139851967498860, −14.846548346330158663215224302344, −14.0331927245778189481822979572, −13.17726561174501857704298333809, −12.849276846215804844308867938653, −11.89981241148620300020507969529, −10.9443920983443466067766235778, −10.31677030969224363396149402326, −9.50663642132489770792103407546, −8.93174194023227941692747328881, −8.09115949055387688425178945665, −6.97128605580726596619804507530, −6.54440932443487500938339464108, −5.43584327927017519518740262589, −4.91394268001021135779200088810, −3.995753073026775523633755787525, −2.8297917106016428549011096441, −2.01560289152749400281879792439, −1.33329705838039402017533546021, 0.06255459757396338898971785291, 1.13329586167176302600770005193, 2.1229674911183510087557407301, 3.01934582048200434287961057870, 3.6321783000475063583101564679, 5.12695110560099919533223061396, 5.51938468183603448460479248845, 6.18515996767149470787803298512, 7.35262682430425933210286788410, 7.824027732508500002444552605435, 8.96992623235024787358237144654, 9.50247772383633406179909000534, 10.46848378827940408505691526811, 10.81584054439881174072369351261, 11.83946964573517565545654862933, 12.74211757910057018588528675682, 13.44627888107659122495277054221, 13.887684996714039213974191465095, 14.79765701667821956557848756656, 15.47792907129843103002669300462, 16.37424300444819406433799178651, 16.956084971483247184412625469485, 17.78511833217148955350992248705, 18.495586813345684027493667825013, 18.781145679658451307584512238598

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