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Properties

Label 1-2268-2268.47-r1-0-0
Degree $1$
Conductor $2268$
Sign $0.949 + 0.313i$
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  + (0.597 + 0.802i)5-s + (−0.993 − 0.116i)11-s + (−0.973 − 0.230i)13-s + (0.766 + 0.642i)17-s + (0.766 − 0.642i)19-s + (−0.835 − 0.549i)23-s + (−0.286 + 0.957i)25-s + (0.286 − 0.957i)29-s + (0.893 − 0.448i)31-s + (0.766 + 0.642i)37-s + (−0.686 + 0.727i)41-s + (−0.597 + 0.802i)43-s + (−0.893 − 0.448i)47-s + (0.5 + 0.866i)53-s + (−0.5 − 0.866i)55-s + ⋯
L(s)  = 1  + (0.597 + 0.802i)5-s + (−0.993 − 0.116i)11-s + (−0.973 − 0.230i)13-s + (0.766 + 0.642i)17-s + (0.766 − 0.642i)19-s + (−0.835 − 0.549i)23-s + (−0.286 + 0.957i)25-s + (0.286 − 0.957i)29-s + (0.893 − 0.448i)31-s + (0.766 + 0.642i)37-s + (−0.686 + 0.727i)41-s + (−0.597 + 0.802i)43-s + (−0.893 − 0.448i)47-s + (0.5 + 0.866i)53-s + (−0.5 − 0.866i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.949 + 0.313i$
Analytic conductor: \(243.730\)
Root analytic conductor: \(243.730\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2268,\ (1:\ ),\ 0.949 + 0.313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.985955179 + 0.3197570055i\)
\(L(\frac12)\) \(\approx\) \(1.985955179 + 0.3197570055i\)
\(L(1)\) \(\approx\) \(1.080134413 + 0.1211009039i\)
\(L(1)\) \(\approx\) \(1.080134413 + 0.1211009039i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.597 + 0.802i)T \)
11 \( 1 + (-0.993 - 0.116i)T \)
13 \( 1 + (-0.973 - 0.230i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.835 - 0.549i)T \)
29 \( 1 + (0.286 - 0.957i)T \)
31 \( 1 + (0.893 - 0.448i)T \)
37 \( 1 + (0.766 + 0.642i)T \)
41 \( 1 + (-0.686 + 0.727i)T \)
43 \( 1 + (-0.597 + 0.802i)T \)
47 \( 1 + (-0.893 - 0.448i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.597 - 0.802i)T \)
61 \( 1 + (0.835 - 0.549i)T \)
67 \( 1 + (-0.973 - 0.230i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (-0.173 - 0.984i)T \)
79 \( 1 + (0.286 - 0.957i)T \)
83 \( 1 + (0.686 + 0.727i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (0.993 + 0.116i)T \)
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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

−19.61155172594562043677948144968, −18.584499577082452885624972514066, −17.995973471140583658976701578578, −17.35961779483200623352080102509, −16.3623724576978687563685433888, −16.1868193973067359771241955587, −15.15250281543834714638658405566, −14.1907064147422137608079059000, −13.756377316863020371715060458786, −12.870370545696391767645984973643, −12.19065027870571332153068993558, −11.68601212716612527875269999543, −10.32247301139574951552643256637, −9.97208881965753886494938313336, −9.24963714995024624557348686095, −8.291795507784857190212520222527, −7.65077230898969177266329811075, −6.82869034222735353743471100603, −5.60368738545624167239449870779, −5.28407481322658695751074929638, −4.482560359036029836839666103841, −3.32397215438839362988240231534, −2.42706556687081127725995400662, −1.56858819762167977853572357890, −0.563921539662022385388953809028, 0.54136275779902731643231868420, 1.82975783261024816236927744520, 2.703583636910567441301590029914, 3.18614467788608830286032656910, 4.48776446132689963839161195038, 5.26383516814786869465953556017, 6.077218416204933356170336149707, 6.74412313315503451360975707171, 7.8238615213364985383860241708, 8.09625818413822460283604104570, 9.571636999822028789551245797643, 9.95851841590089984978265311641, 10.56829049447523151854547766088, 11.510014959393889563507290251220, 12.18961481464547365932068277641, 13.22070387480121338894673869213, 13.629403384077063535307785465775, 14.59060354255952658150738908149, 15.064862094634615785014145057084, 15.85502673264339913840935767210, 16.78421072409924285967246338495, 17.426706821170919542574149883208, 18.18102894051504810375535686931, 18.65882459516953824569264389852, 19.47596422178596925252428790072

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