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Properties

Label 1-1148-1148.51-r1-0-0
Degree $1$
Conductor $1148$
Sign $0.508 + 0.861i$
Analytic cond. $123.369$
Root an. cond. $123.369$
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)3-s + (0.913 + 0.406i)5-s + (−0.5 − 0.866i)9-s + (−0.913 + 0.406i)11-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)15-s + (0.913 − 0.406i)17-s + (0.978 − 0.207i)19-s + (−0.669 + 0.743i)23-s + (0.669 + 0.743i)25-s − 27-s + (−0.809 + 0.587i)29-s + (−0.913 + 0.406i)31-s + (−0.104 + 0.994i)33-s + (0.913 + 0.406i)37-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (0.913 + 0.406i)5-s + (−0.5 − 0.866i)9-s + (−0.913 + 0.406i)11-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)15-s + (0.913 − 0.406i)17-s + (0.978 − 0.207i)19-s + (−0.669 + 0.743i)23-s + (0.669 + 0.743i)25-s − 27-s + (−0.809 + 0.587i)29-s + (−0.913 + 0.406i)31-s + (−0.104 + 0.994i)33-s + (0.913 + 0.406i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.508 + 0.861i$
Analytic conductor: \(123.369\)
Root analytic conductor: \(123.369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (1:\ ),\ 0.508 + 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.042451768 + 1.166191909i\)
\(L(\frac12)\) \(\approx\) \(2.042451768 + 1.166191909i\)
\(L(1)\) \(\approx\) \(1.354798883 - 0.05085718135i\)
\(L(1)\) \(\approx\) \(1.354798883 - 0.05085718135i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.913 + 0.406i)T \)
11 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (0.978 - 0.207i)T \)
23 \( 1 + (-0.669 + 0.743i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (-0.669 + 0.743i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
61 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (0.104 + 0.994i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.978 + 0.207i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−20.90241589756926272354817048811, −20.45494951593505024919541204685, −19.70994441906530986126675735207, −18.46412613795583954172961251129, −18.05367366592200646065938571358, −16.772641942628743258207391431737, −16.46555394075561310925230990940, −15.556253110385459890425766288514, −14.74365203587082366790084223752, −13.98715830194715508414090297939, −13.25173417691310971017873978079, −12.59152468259916395784032784666, −11.28759429479918585951618248498, −10.43201986168153597177519918162, −9.90963170627718633868297464936, −9.1725636340717509732057350293, −8.15268497770712576495774651645, −7.7162720172935422485442483965, −5.94036315356016376663699078106, −5.58802678091896052252397309373, −4.69405150357910268747857438297, −3.53659225652045785771941209808, −2.79520251705137324611912844052, −1.77485494636826064241457753459, −0.4141739563996395260718709462, 1.191193072049492181559843124948, 1.94901289757675108130529988888, 2.82032928352156210078117168878, 3.67465287199899315358966633084, 5.227503252528693905023018011513, 5.82695434963806051046106253557, 6.95191326422413447092730979548, 7.38697529911142669318098194755, 8.3841113175545432164290338185, 9.456229756958233374468761683836, 9.839802111804319815175187315959, 11.08451162160206475840968957931, 11.85574168788598910495504087470, 12.84778270554948949375910511583, 13.4403967156372861264675424751, 14.17450526437791050316775712114, 14.67117377997602024602729668323, 15.79749548840117667224455238464, 16.68377593978884390813368282401, 17.64908261066870436037443193549, 18.42471374051028673219706975747, 18.53211789032916152560908412671, 19.681738705288264483483235648, 20.51449402548385568253370308349, 21.09040026493244912704605261521

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