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Properties

Label 1-2268-2268.367-r0-0-0
Degree $1$
Conductor $2268$
Sign $-0.221 + 0.975i$
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  + (0.286 + 0.957i)5-s + (−0.973 + 0.230i)11-s + (0.0581 − 0.998i)13-s + (−0.766 − 0.642i)17-s + (0.766 − 0.642i)19-s + (−0.396 + 0.918i)23-s + (−0.835 + 0.549i)25-s + (0.893 + 0.448i)29-s + (0.396 − 0.918i)31-s + (−0.939 + 0.342i)37-s + (0.835 + 0.549i)41-s + (−0.973 + 0.230i)43-s + (0.396 + 0.918i)47-s + 53-s + (−0.5 − 0.866i)55-s + ⋯
L(s)  = 1  + (0.286 + 0.957i)5-s + (−0.973 + 0.230i)11-s + (0.0581 − 0.998i)13-s + (−0.766 − 0.642i)17-s + (0.766 − 0.642i)19-s + (−0.396 + 0.918i)23-s + (−0.835 + 0.549i)25-s + (0.893 + 0.448i)29-s + (0.396 − 0.918i)31-s + (−0.939 + 0.342i)37-s + (0.835 + 0.549i)41-s + (−0.973 + 0.230i)43-s + (0.396 + 0.918i)47-s + 53-s + (−0.5 − 0.866i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.221 + 0.975i$
Analytic conductor: \(10.5325\)
Root analytic conductor: \(10.5325\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2268,\ (0:\ ),\ -0.221 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7284978190 + 0.9128645502i\)
\(L(\frac12)\) \(\approx\) \(0.7284978190 + 0.9128645502i\)
\(L(1)\) \(\approx\) \(0.9543345849 + 0.2287182331i\)
\(L(1)\) \(\approx\) \(0.9543345849 + 0.2287182331i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.286 + 0.957i)T \)
11 \( 1 + (-0.973 + 0.230i)T \)
13 \( 1 + (0.0581 - 0.998i)T \)
17 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.396 + 0.918i)T \)
29 \( 1 + (0.893 + 0.448i)T \)
31 \( 1 + (0.396 - 0.918i)T \)
37 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (0.835 + 0.549i)T \)
43 \( 1 + (-0.973 + 0.230i)T \)
47 \( 1 + (0.396 + 0.918i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.686 + 0.727i)T \)
61 \( 1 + (0.993 + 0.116i)T \)
67 \( 1 + (0.835 + 0.549i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (-0.766 + 0.642i)T \)
79 \( 1 + (0.0581 + 0.998i)T \)
83 \( 1 + (-0.835 + 0.549i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
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

−19.519923623350695192449916138366, −18.732669059943328796415331518557, −17.97881980531366910171006158240, −17.32111676619612375085355459274, −16.4583542300112576114988051153, −16.02783535218667107977777744296, −15.32681885381812632487799905671, −14.12634360185921510371534045264, −13.76078531098381800149275594271, −12.89620077708298778637503917697, −12.2460019388605987186570393983, −11.59311514294441594078414673703, −10.468936350901200187892032504933, −10.04573644057937961962792214855, −8.8943994360292926383389694579, −8.567333498869897919356126834114, −7.69973577533903331008861496952, −6.67354066249977271388918792287, −5.91740932500905471277320895949, −5.06345287102551547869538453635, −4.43646096256098862097501871545, −3.51689318067453647616307987257, −2.30567136106468787566021476896, −1.65220284389236866407456249679, −0.40885919380824204778926544374, 1.075724464451848776015310189729, 2.497289170127970691376464910966, 2.77076156375577585588346396309, 3.7895472912494581416353218369, 4.95733318173670849565523053376, 5.56109443274247566138100250513, 6.47612718552410331369996293238, 7.298238506373933622822300378082, 7.821117313277682454829138519728, 8.81819822175495604613085150603, 9.87386700659480140064162951616, 10.20920016450710944157978947781, 11.15240776417752726541360979793, 11.64394485860272834826865171034, 12.77560702562907565357284200965, 13.45984338252686263910104132143, 13.96176373137768632443251550358, 14.94813642288434807467582831608, 15.60942633089339466318995563534, 15.95246874649013088981718773075, 17.30605152259910617707232847726, 17.870806075167453535192114074719, 18.230471751639781908871412837606, 19.08542691395522327868123175429, 19.94567337773491649723277014731

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