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Properties

Label 1-1057-1057.16-r0-0-0
Degree $1$
Conductor $1057$
Sign $-0.949 + 0.313i$
Analytic cond. $4.90868$
Root an. cond. $4.90868$
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  + (−0.5 + 0.866i)2-s + (−0.978 − 0.207i)3-s + (−0.5 − 0.866i)4-s + (−0.104 − 0.994i)5-s + (0.669 − 0.743i)6-s + 8-s + (0.913 + 0.406i)9-s + (0.913 + 0.406i)10-s + (0.309 − 0.951i)11-s + (0.309 + 0.951i)12-s + (−0.978 − 0.207i)13-s + (−0.104 + 0.994i)15-s + (−0.5 + 0.866i)16-s + (−0.809 − 0.587i)17-s + (−0.809 + 0.587i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.978 − 0.207i)3-s + (−0.5 − 0.866i)4-s + (−0.104 − 0.994i)5-s + (0.669 − 0.743i)6-s + 8-s + (0.913 + 0.406i)9-s + (0.913 + 0.406i)10-s + (0.309 − 0.951i)11-s + (0.309 + 0.951i)12-s + (−0.978 − 0.207i)13-s + (−0.104 + 0.994i)15-s + (−0.5 + 0.866i)16-s + (−0.809 − 0.587i)17-s + (−0.809 + 0.587i)18-s + (−0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1057\)    =    \(7 \cdot 151\)
Sign: $-0.949 + 0.313i$
Analytic conductor: \(4.90868\)
Root analytic conductor: \(4.90868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1057} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1057,\ (0:\ ),\ -0.949 + 0.313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01549835115 - 0.09629324507i\)
\(L(\frac12)\) \(\approx\) \(0.01549835115 - 0.09629324507i\)
\(L(1)\) \(\approx\) \(0.4557665948 - 0.03399826892i\)
\(L(1)\) \(\approx\) \(0.4557665948 - 0.03399826892i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
151 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.978 - 0.207i)T \)
5 \( 1 + (-0.104 - 0.994i)T \)
11 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (-0.978 - 0.207i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.913 - 0.406i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (0.669 + 0.743i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.913 - 0.406i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (-0.104 - 0.994i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.104 + 0.994i)T \)
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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

−22.01916650359658591160369803890, −21.32078752699475940140359819384, −20.32602776556406215539882483907, −19.52129424613827559863261246345, −18.77978183974113253045051028547, −17.99733471453388517405110473291, −17.50581046033910914804909717418, −16.78771122250760931142624180334, −15.853741608620955967978319360993, −14.83592388355805324267863545029, −14.141949294476679389374575025628, −12.68316333567066518721918810790, −12.3597179072102482619102858963, −11.52749252156915506659235178759, −10.597702967402537319567630167153, −10.305125958793400313707300441167, −9.46569403641725293243418816066, −8.356821865844437028756037775641, −7.06796137290880464122115533630, −6.84691114989831727832316914745, −5.446518509262736284589321495741, −4.32924279931375972484745527736, −3.791093021505776634227707402570, −2.406626279141229431276765316, −1.677623896777757530555835498374, 0.07008316589863870831856234309, 0.91746260507350589474215283774, 2.13620379792131241683193713900, 4.1261628311199329809116855936, 4.835901444391174788554328122727, 5.60007277694842219597875062675, 6.30081424829522887685885514382, 7.30172004527296879722451413390, 7.97308097010225167285032702139, 9.06936330099071939780349588982, 9.57218933771355087972239964325, 10.72955736965574638414089340362, 11.45580502217264477676441238626, 12.36035885822280373940679118846, 13.31970138042243297104597174336, 13.84807855027803348786080651524, 15.205532572848166831478088960098, 15.84619572926324568256514263936, 16.46892625309996211134683079883, 17.35319097403204112552263254152, 17.47193780150560699769660278859, 18.59528005607696640359404328597, 19.4496100038947134756268571456, 19.93783987121107396125043737219, 21.29218017970854125527998920997

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