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Properties

Label 1-1063-1063.120-r0-0-0
Degree $1$
Conductor $1063$
Sign $0.949 - 0.314i$
Analytic cond. $4.93655$
Root an. cond. $4.93655$
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  + (0.525 + 0.850i)2-s + (0.565 − 0.825i)3-s + (−0.447 + 0.894i)4-s + (0.999 − 0.0354i)5-s + (0.998 + 0.0473i)6-s + (0.173 − 0.984i)7-s + (−0.996 + 0.0886i)8-s + (−0.361 − 0.932i)9-s + (0.555 + 0.831i)10-s + (0.718 + 0.695i)11-s + (0.484 + 0.874i)12-s + (0.150 − 0.988i)13-s + (0.929 − 0.369i)14-s + (0.535 − 0.844i)15-s + (−0.598 − 0.800i)16-s + (0.994 + 0.106i)17-s + ⋯
L(s)  = 1  + (0.525 + 0.850i)2-s + (0.565 − 0.825i)3-s + (−0.447 + 0.894i)4-s + (0.999 − 0.0354i)5-s + (0.998 + 0.0473i)6-s + (0.173 − 0.984i)7-s + (−0.996 + 0.0886i)8-s + (−0.361 − 0.932i)9-s + (0.555 + 0.831i)10-s + (0.718 + 0.695i)11-s + (0.484 + 0.874i)12-s + (0.150 − 0.988i)13-s + (0.929 − 0.369i)14-s + (0.535 − 0.844i)15-s + (−0.598 − 0.800i)16-s + (0.994 + 0.106i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1063 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1063\)
Sign: $0.949 - 0.314i$
Analytic conductor: \(4.93655\)
Root analytic conductor: \(4.93655\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1063} (120, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1063,\ (0:\ ),\ 0.949 - 0.314i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.774553689 - 0.4480163702i\)
\(L(\frac12)\) \(\approx\) \(2.774553689 - 0.4480163702i\)
\(L(1)\) \(\approx\) \(1.864208974 + 0.06455377816i\)
\(L(1)\) \(\approx\) \(1.864208974 + 0.06455377816i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1063 \( 1 \)
good2 \( 1 + (0.525 + 0.850i)T \)
3 \( 1 + (0.565 - 0.825i)T \)
5 \( 1 + (0.999 - 0.0354i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.718 + 0.695i)T \)
13 \( 1 + (0.150 - 0.988i)T \)
17 \( 1 + (0.994 + 0.106i)T \)
19 \( 1 + (-0.0856 - 0.996i)T \)
23 \( 1 + (-0.864 + 0.502i)T \)
29 \( 1 + (-0.179 - 0.983i)T \)
31 \( 1 + (-0.997 - 0.0768i)T \)
37 \( 1 + (-0.917 + 0.396i)T \)
41 \( 1 + (0.842 - 0.537i)T \)
43 \( 1 + (0.421 + 0.906i)T \)
47 \( 1 + (-0.922 + 0.386i)T \)
53 \( 1 + (-0.985 - 0.170i)T \)
59 \( 1 + (0.603 + 0.797i)T \)
61 \( 1 + (0.873 - 0.487i)T \)
67 \( 1 + (0.963 + 0.268i)T \)
71 \( 1 + (-0.437 + 0.899i)T \)
73 \( 1 + (0.982 - 0.188i)T \)
79 \( 1 + (0.0561 + 0.998i)T \)
83 \( 1 + (0.701 - 0.712i)T \)
89 \( 1 + (0.453 - 0.891i)T \)
97 \( 1 + (0.300 + 0.953i)T \)
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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

−21.45029158669063106979534663573, −20.99799545515536199565930929240, −20.28904742756639379537474060237, −19.17299962457532418875791463280, −18.76882204974845240093554955794, −17.89643636352830615402528201008, −16.62149686922020466128056092379, −16.10867781603769539774624094292, −14.79617212193550373163227983348, −14.260555273519834867811974126603, −14.037847406491768177268624309908, −12.79863806807913329736351467678, −12.03294268902406520969415037129, −11.13553736794962323344359098083, −10.36123652244905047882138594748, −9.47909106568327966486884597085, −9.087509730733816660075059637352, −8.26178001572583918515573469374, −6.47814142784378060317339852381, −5.65422207011665188139187967700, −5.10253784062541216388532664375, −3.896085988560239411343552098174, −3.25462214381047852111213299305, −2.14606394819549297641414599028, −1.60749114256606496134869704971, 0.942353570361369004268135919209, 2.07905800394242956134392466655, 3.20417295948727123923454240223, 4.042673820097699124818363519757, 5.21975348465869186645171092568, 6.07296415615668414883569233381, 6.841882070801813087304214085457, 7.55604590555444927009193394702, 8.2373709003437956630513120397, 9.34423787792393398236893158964, 9.93224064017946517282996605532, 11.31560736921408966287076877691, 12.4395035794940622783450416104, 13.0139152870323961289699817529, 13.666498244863697690843597952625, 14.35149787782699830992476557265, 14.81760611413432929682260166880, 15.9235163058562213722047011328, 17.02919070298268221658705803673, 17.6207108294503753713659442159, 17.84616409859695962828743033317, 19.078913511603751541076611674398, 20.1152435450020088066892622218, 20.65258456645488447673139699941, 21.46059673954151473302693448909

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