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Properties

Label 1-1057-1057.991-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: $\chi_{1057} (991, \cdot )$
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

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

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