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Properties

Label 1-1248-1248.251-r0-0-0
Degree $1$
Conductor $1248$
Sign $0.0526 + 0.998i$
Analytic cond. $5.79568$
Root an. cond. $5.79568$
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  + (0.707 + 0.707i)5-s + (−0.866 − 0.5i)7-s + (0.258 − 0.965i)11-s + (−0.5 + 0.866i)17-s + (0.258 + 0.965i)19-s + (−0.866 + 0.5i)23-s + i·25-s + (−0.965 − 0.258i)29-s + 31-s + (−0.258 − 0.965i)35-s + (−0.258 + 0.965i)37-s + (−0.866 + 0.5i)41-s + (−0.965 + 0.258i)43-s + 47-s + (0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)5-s + (−0.866 − 0.5i)7-s + (0.258 − 0.965i)11-s + (−0.5 + 0.866i)17-s + (0.258 + 0.965i)19-s + (−0.866 + 0.5i)23-s + i·25-s + (−0.965 − 0.258i)29-s + 31-s + (−0.258 − 0.965i)35-s + (−0.258 + 0.965i)37-s + (−0.866 + 0.5i)41-s + (−0.965 + 0.258i)43-s + 47-s + (0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0526 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0526 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $0.0526 + 0.998i$
Analytic conductor: \(5.79568\)
Root analytic conductor: \(5.79568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1248,\ (0:\ ),\ 0.0526 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8778843294 + 0.8328590017i\)
\(L(\frac12)\) \(\approx\) \(0.8778843294 + 0.8328590017i\)
\(L(1)\) \(\approx\) \(0.9958615982 + 0.2018855674i\)
\(L(1)\) \(\approx\) \(0.9958615982 + 0.2018855674i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.258 - 0.965i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.258 + 0.965i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.965 - 0.258i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.258 + 0.965i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (-0.965 + 0.258i)T \)
47 \( 1 + T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (0.965 - 0.258i)T \)
61 \( 1 + (-0.258 - 0.965i)T \)
67 \( 1 + (0.965 + 0.258i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−20.77289416047495793420768450993, −20.17053366518275718855191413548, −19.55866594090104670419318557451, −18.44960174410855142091508725675, −17.86273469972308559283673166108, −17.09779657281664288710359807652, −16.26249965412591479044093084276, −15.64276543552843329262136438393, −14.80949099225119130228197574579, −13.67877383741155049049155068188, −13.25550643050169970521376377810, −12.29405324368586300376203375531, −11.858764968817311394824430290963, −10.53597821353062927735058529414, −9.6876719007727295569578694306, −9.22677782917730798061310857751, −8.46563914974676079594392029439, −7.154650701045665246003878922516, −6.54568906481750360646813041844, −5.53301205531230576882237470051, −4.84721571288835736126942036564, −3.8488907469729985970139249265, −2.56077511879244424794018729595, −1.95341754816497693879634930700, −0.48569970102459205666050431619, 1.23710875955035736189601715259, 2.320915120816851571826673150610, 3.41179961738944200379021284334, 3.88672105719273494739984735992, 5.40525559911597247301226494674, 6.224114275901614501706802085353, 6.63560665402830078767482950999, 7.76676345100838931949642113004, 8.64503027193129623563613231597, 9.771322554020597884575105815304, 10.136081288225120557115264256361, 11.02524282104866988543747353137, 11.85654900090204173888096389478, 12.959549259668091526277579994069, 13.62658147619643745312958184922, 14.1244410402690849998066261588, 15.12002608053351806683589196141, 15.895324696299063364242725752392, 16.923293911821979834767625314489, 17.19146385211417661075053113360, 18.45008427646706781758791004804, 18.84229350472451207900568550685, 19.70215040288251357589766543394, 20.45252912918691539512652369988, 21.48900260152015268892684597169

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