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Properties

Label 1-2268-2268.535-r0-0-0
Degree $1$
Conductor $2268$
Sign $0.0297 - 0.999i$
Analytic cond. $10.5325$
Root an. cond. $10.5325$
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

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

Invariants

Degree: \(1\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.0297 - 0.999i$
Analytic conductor: \(10.5325\)
Root analytic conductor: \(10.5325\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2268,\ (0:\ ),\ 0.0297 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6615992481 - 0.6421795485i\)
\(L(\frac12)\) \(\approx\) \(0.6615992481 - 0.6421795485i\)
\(L(1)\) \(\approx\) \(0.8525060491 - 0.08749921486i\)
\(L(1)\) \(\approx\) \(0.8525060491 - 0.08749921486i\)

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

−19.838928213383871875981221954120, −19.38150232258314539206022452548, −18.27864567156893063168354915868, −17.83949848028857464607653472516, −16.83640635491947250164605604432, −16.27097165454954960735883263418, −15.54177557510559846992539771808, −14.65352143251966571033444004876, −14.47057173381049620287497146250, −13.001607490165713307109365763851, −12.58120620500290787459295171780, −11.97078829263671117133625798555, −11.083440669585291541454535055298, −10.36406970430796652985640585694, −9.60936307965784123373726195434, −8.51061936654240819166521866814, −8.126720066507799096712980473865, −7.270922188060452359417408173856, −6.451545313829859463752240462578, −5.55796445783806518025836539171, −4.58651629535025658863598433960, −3.90488950295180243911406681122, −3.21699015157973792206405553047, −1.98285600295416917678228860311, −1.02710043729977851832716882254, 0.35829631444502182778170906001, 1.51733285202240122754555833187, 2.72714148089039057574811331390, 3.48597431327158279684205308674, 4.28561766477481703503898075797, 4.9918328508239747818937792193, 6.16986369620293134620906801575, 6.87153473673462552253797625285, 7.51427680591923852208339566292, 8.44652353114214798962854255877, 9.10065266457840715204288359486, 9.842169158458414184446566769798, 11.01232243233895234445574561467, 11.56154577891914478740408823778, 11.86491557365265946622811295184, 12.98887810909820938746726497733, 13.90290211926596703774797876685, 14.28755165251635061252682733013, 15.32416026628780704451563441350, 15.89820672075628660382406349504, 16.48573977014973840633714551362, 17.28303587090859789571569762693, 18.23648442527692855701222109195, 18.79741325451330221928410596026, 19.6369734914863098727625919771

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