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Properties

Label 1-1287-1287.448-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.555 - 0.831i$
Analytic cond. $5.97680$
Root an. cond. $5.97680$
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  + (−0.743 + 0.669i)2-s + (0.104 − 0.994i)4-s + (−0.207 + 0.978i)5-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.5 − 0.866i)10-s + (−0.104 − 0.994i)14-s + (−0.978 − 0.207i)16-s + (−0.978 − 0.207i)17-s + (0.994 − 0.104i)19-s + (0.951 + 0.309i)20-s − 23-s + (−0.913 − 0.406i)25-s + (0.743 + 0.669i)28-s + (−0.913 + 0.406i)29-s + ⋯
L(s)  = 1  + (−0.743 + 0.669i)2-s + (0.104 − 0.994i)4-s + (−0.207 + 0.978i)5-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.5 − 0.866i)10-s + (−0.104 − 0.994i)14-s + (−0.978 − 0.207i)16-s + (−0.978 − 0.207i)17-s + (0.994 − 0.104i)19-s + (0.951 + 0.309i)20-s − 23-s + (−0.913 − 0.406i)25-s + (0.743 + 0.669i)28-s + (−0.913 + 0.406i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.555 - 0.831i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (448, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.555 - 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1901975982 - 0.1016167011i\)
\(L(\frac12)\) \(\approx\) \(0.1901975982 - 0.1016167011i\)
\(L(1)\) \(\approx\) \(0.4923549966 + 0.2336312162i\)
\(L(1)\) \(\approx\) \(0.4923549966 + 0.2336312162i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.743 + 0.669i)T \)
5 \( 1 + (-0.207 + 0.978i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (0.994 - 0.104i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.913 + 0.406i)T \)
31 \( 1 + (0.743 - 0.669i)T \)
37 \( 1 + (-0.994 - 0.104i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.406 + 0.913i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.994 - 0.104i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.207 + 0.978i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (0.743 + 0.669i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.951 - 0.309i)T \)
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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

−20.7458536352260473045844224591, −20.28774920036590315038749539504, −19.705170363602349527480852319374, −19.08217266048259091507206054700, −17.97473563901988048661744622792, −17.42131108281092697720480823379, −16.557036061803627834704170591140, −16.13742869799354000082041916346, −15.31171514981119464416393932763, −13.61175464810691970442680693009, −13.49683693011205494226918254507, −12.38175684794971584267903823273, −11.89256704124829358336222029979, −10.91201156588575400786462739127, −10.10741337872777951841638661881, −9.41255742096209446555042354668, −8.68097256424257069007401617723, −7.82838323030236672621378792428, −7.12727473348124033911097199799, −6.0552737007539946479708772415, −4.725050287355996976683484807942, −3.99263083423714240155670010759, −3.187402907894246120562577677750, −1.91971100878188871788453045183, −1.00786522980035115677736877452, 0.124103442607204429351491189174, 1.873527821952393943039602355078, 2.67296341570732247851476007487, 3.76896045421533623093493515919, 5.08466744343589613705914158081, 5.979669520642765181487615813609, 6.606262629821678092480098657945, 7.38113310438942741196610825574, 8.18600545357829234819476419600, 9.18183630688482887494688182461, 9.72605182337294708598009034829, 10.631800573212221185048537963721, 11.385859586009281234534081180729, 12.16676876829165136740576443172, 13.45588698185030385778739037204, 14.14741965431666399272439880473, 14.98317363485984847831013670383, 15.7574614035365568167110573218, 15.968447270502346123492816268031, 17.22204832736317557356590407447, 17.95657628182274889907702237488, 18.52698531071120366940430704261, 19.13987653529560917950821220613, 19.83594919209956591444009634468, 20.67337184835613226436307071181

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