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Properties

Label 1-5824-5824.1005-r0-0-0
Degree $1$
Conductor $5824$
Sign $-0.241 + 0.970i$
Analytic cond. $27.0465$
Root an. cond. $27.0465$
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  + (−0.608 − 0.793i)3-s + (−0.130 + 0.991i)5-s + (−0.258 + 0.965i)9-s + (0.991 + 0.130i)11-s + (0.866 − 0.5i)15-s + i·17-s + (0.130 + 0.991i)19-s + (0.707 + 0.707i)23-s + (−0.965 − 0.258i)25-s + (0.923 − 0.382i)27-s + (−0.991 + 0.130i)29-s + (−0.5 − 0.866i)31-s + (−0.5 − 0.866i)33-s + (−0.923 + 0.382i)37-s + (−0.258 + 0.965i)41-s + ⋯
L(s)  = 1  + (−0.608 − 0.793i)3-s + (−0.130 + 0.991i)5-s + (−0.258 + 0.965i)9-s + (0.991 + 0.130i)11-s + (0.866 − 0.5i)15-s + i·17-s + (0.130 + 0.991i)19-s + (0.707 + 0.707i)23-s + (−0.965 − 0.258i)25-s + (0.923 − 0.382i)27-s + (−0.991 + 0.130i)29-s + (−0.5 − 0.866i)31-s + (−0.5 − 0.866i)33-s + (−0.923 + 0.382i)37-s + (−0.258 + 0.965i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5824 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5824 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5824\)    =    \(2^{6} \cdot 7 \cdot 13\)
Sign: $-0.241 + 0.970i$
Analytic conductor: \(27.0465\)
Root analytic conductor: \(27.0465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5824} (1005, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5824,\ (0:\ ),\ -0.241 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7428808788 + 0.9504784264i\)
\(L(\frac12)\) \(\approx\) \(0.7428808788 + 0.9504784264i\)
\(L(1)\) \(\approx\) \(0.8718672523 + 0.1478716953i\)
\(L(1)\) \(\approx\) \(0.8718672523 + 0.1478716953i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.608 - 0.793i)T \)
5 \( 1 + (-0.130 + 0.991i)T \)
11 \( 1 + (0.991 + 0.130i)T \)
17 \( 1 + iT \)
19 \( 1 + (0.130 + 0.991i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (-0.991 + 0.130i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.923 + 0.382i)T \)
41 \( 1 + (-0.258 + 0.965i)T \)
43 \( 1 + (0.991 + 0.130i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.608 - 0.793i)T \)
59 \( 1 + (0.923 - 0.382i)T \)
61 \( 1 + (0.991 - 0.130i)T \)
67 \( 1 + (0.608 + 0.793i)T \)
71 \( 1 + (0.258 + 0.965i)T \)
73 \( 1 + (0.258 - 0.965i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (-0.923 - 0.382i)T \)
89 \( 1 + (-0.707 + 0.707i)T \)
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

−17.43002068220975192622578428611, −16.74034511625112381149119866431, −16.47807127616069810801759040455, −15.60288178235838508469197537521, −15.270820285589289844798668987953, −14.24891318313273780214894643862, −13.76141788037820857023700013409, −12.70518367736554394044047649787, −12.29429825966884648435731799349, −11.527607354847569209583021708216, −11.09799304234843755132184023771, −10.26474393419749393151733432484, −9.37592197495867287118885003706, −9.0131203786315805570778020676, −8.569098965616337447334151930011, −7.20083554121675911260251031088, −6.873377854932445438531059690628, −5.697502110395048875930813307511, −5.32235415069121214705795980546, −4.5660688533353188519281342684, −3.988553898214848233229452411679, −3.26009107309085398103833250722, −2.16592754116846453512698956340, −1.02001862288779098456620154699, −0.41744302162612982953663367971, 1.07042743966664677468999301871, 1.7863476691342090961982089105, 2.49278683816077717505664910941, 3.63250347225162001573962288861, 3.97954993704643768763659549077, 5.28663010346085890827397227495, 5.85039568030850231138837773834, 6.533906710966613607260816911340, 7.0578697432473175561645977158, 7.72287965524392701025141693257, 8.36628344028493164621548550831, 9.37486995317690302433145504842, 10.103370693716192917711549269276, 10.84482744394890837312728318269, 11.39500277915558020990867210741, 11.88807664055168959443520964422, 12.69455647866189948018768682187, 13.24347532158378812241980120783, 14.107249071964084600264435578374, 14.60987855719393243861060318141, 15.18259902702581808339591805236, 16.103645852728759775775962833210, 16.898097194444365388968692695003, 17.3113905617920955768216249126, 17.92914297339336065614082342147

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