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Properties

Label 1-1148-1148.1031-r0-0-0
Degree $1$
Conductor $1148$
Sign $-0.739 + 0.673i$
Analytic cond. $5.33128$
Root an. cond. $5.33128$
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  + (0.258 + 0.965i)3-s + (0.207 + 0.978i)5-s + (−0.866 + 0.5i)9-s + (0.838 − 0.544i)11-s + (0.156 − 0.987i)13-s + (−0.891 + 0.453i)15-s + (0.544 + 0.838i)17-s + (−0.777 + 0.629i)19-s + (0.913 + 0.406i)23-s + (−0.913 + 0.406i)25-s + (−0.707 − 0.707i)27-s + (0.453 + 0.891i)29-s + (−0.978 − 0.207i)31-s + (0.743 + 0.669i)33-s + (−0.978 + 0.207i)37-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)3-s + (0.207 + 0.978i)5-s + (−0.866 + 0.5i)9-s + (0.838 − 0.544i)11-s + (0.156 − 0.987i)13-s + (−0.891 + 0.453i)15-s + (0.544 + 0.838i)17-s + (−0.777 + 0.629i)19-s + (0.913 + 0.406i)23-s + (−0.913 + 0.406i)25-s + (−0.707 − 0.707i)27-s + (0.453 + 0.891i)29-s + (−0.978 − 0.207i)31-s + (0.743 + 0.669i)33-s + (−0.978 + 0.207i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.739 + 0.673i$
Analytic conductor: \(5.33128\)
Root analytic conductor: \(5.33128\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (1031, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (0:\ ),\ -0.739 + 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5678477969 + 1.467047260i\)
\(L(\frac12)\) \(\approx\) \(0.5678477969 + 1.467047260i\)
\(L(1)\) \(\approx\) \(0.9886700773 + 0.6492222095i\)
\(L(1)\) \(\approx\) \(0.9886700773 + 0.6492222095i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (0.207 + 0.978i)T \)
11 \( 1 + (0.838 - 0.544i)T \)
13 \( 1 + (0.156 - 0.987i)T \)
17 \( 1 + (0.544 + 0.838i)T \)
19 \( 1 + (-0.777 + 0.629i)T \)
23 \( 1 + (0.913 + 0.406i)T \)
29 \( 1 + (0.453 + 0.891i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (0.587 + 0.809i)T \)
47 \( 1 + (0.358 + 0.933i)T \)
53 \( 1 + (0.998 + 0.0523i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (-0.994 - 0.104i)T \)
67 \( 1 + (-0.0523 + 0.998i)T \)
71 \( 1 + (0.891 + 0.453i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.965 + 0.258i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.629 - 0.777i)T \)
97 \( 1 + (0.891 - 0.453i)T \)
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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

−20.8689733905396911217578450224, −20.17084742469032971475251137586, −19.446119082298241890627768556420, −18.817220087617521209715138270714, −17.89453649639393447166812775438, −17.063338335697806923643068314730, −16.71022086482619151526321013226, −15.544852669661896649208297460358, −14.5717958907628683455497842651, −13.84562308971223435178419862298, −13.22497533431564715938870492043, −12.23241231407901368507356227304, −11.981422200346829590111135202648, −10.89716370906265632037662182184, −9.47621207419877393839500048425, −9.04232322595173716445879395744, −8.31345593970643421285656485191, −7.16139628338702733557941211553, −6.68336779918595295625142551667, −5.60395044585853698261316032075, −4.67135191807911908594038632444, −3.72438084998269496478354527082, −2.369177400466226815693311196960, −1.64826528090739780910922545072, −0.63047217826160525660158358962, 1.46654111363243621751717793983, 2.813147685668241674700993872549, 3.43217273235676080365061999605, 4.16146401564021745773908810581, 5.51360615144795471164107668542, 6.01458178850336020114649119655, 7.125245024593298656974513609889, 8.15389090540318964199278061860, 8.90774578685273448752368083625, 9.80848951485216459890695718674, 10.70920810478840634363847226067, 10.88488894457495052843517503110, 12.06633160412357942752743903491, 13.1154171270190384347035896316, 14.10569972739559808510024074504, 14.70478467469990856267962976590, 15.163499545419626369986611062731, 16.129030059360292832552018501598, 16.97904731087478048748125506894, 17.55630475232272884075217768159, 18.64812526708730034089234985446, 19.37268156211618783303053071939, 19.98213424629110699878056476244, 21.12710925889446030334214002541, 21.46505410083665916940718980319

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