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Properties

Label 1-1536-1536.587-r0-0-0
Degree $1$
Conductor $1536$
Sign $0.254 - 0.966i$
Analytic cond. $7.13315$
Root an. cond. $7.13315$
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  + (0.242 − 0.970i)5-s + (0.773 − 0.634i)7-s + (−0.903 − 0.427i)11-s + (0.857 + 0.514i)13-s + (−0.831 − 0.555i)17-s + (0.595 + 0.803i)19-s + (0.956 + 0.290i)23-s + (−0.881 − 0.471i)25-s + (0.941 + 0.336i)29-s + (0.382 − 0.923i)31-s + (−0.427 − 0.903i)35-s + (0.146 + 0.989i)37-s + (0.881 − 0.471i)41-s + (0.671 + 0.740i)43-s + (−0.195 − 0.980i)47-s + ⋯
L(s)  = 1  + (0.242 − 0.970i)5-s + (0.773 − 0.634i)7-s + (−0.903 − 0.427i)11-s + (0.857 + 0.514i)13-s + (−0.831 − 0.555i)17-s + (0.595 + 0.803i)19-s + (0.956 + 0.290i)23-s + (−0.881 − 0.471i)25-s + (0.941 + 0.336i)29-s + (0.382 − 0.923i)31-s + (−0.427 − 0.903i)35-s + (0.146 + 0.989i)37-s + (0.881 − 0.471i)41-s + (0.671 + 0.740i)43-s + (−0.195 − 0.980i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $0.254 - 0.966i$
Analytic conductor: \(7.13315\)
Root analytic conductor: \(7.13315\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1536} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1536,\ (0:\ ),\ 0.254 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.437230759 - 1.107503673i\)
\(L(\frac12)\) \(\approx\) \(1.437230759 - 1.107503673i\)
\(L(1)\) \(\approx\) \(1.179160688 - 0.3656125542i\)
\(L(1)\) \(\approx\) \(1.179160688 - 0.3656125542i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.242 - 0.970i)T \)
7 \( 1 + (0.773 - 0.634i)T \)
11 \( 1 + (-0.903 - 0.427i)T \)
13 \( 1 + (0.857 + 0.514i)T \)
17 \( 1 + (-0.831 - 0.555i)T \)
19 \( 1 + (0.595 + 0.803i)T \)
23 \( 1 + (0.956 + 0.290i)T \)
29 \( 1 + (0.941 + 0.336i)T \)
31 \( 1 + (0.382 - 0.923i)T \)
37 \( 1 + (0.146 + 0.989i)T \)
41 \( 1 + (0.881 - 0.471i)T \)
43 \( 1 + (0.671 + 0.740i)T \)
47 \( 1 + (-0.195 - 0.980i)T \)
53 \( 1 + (0.941 - 0.336i)T \)
59 \( 1 + (-0.857 + 0.514i)T \)
61 \( 1 + (0.998 - 0.0490i)T \)
67 \( 1 + (-0.0490 - 0.998i)T \)
71 \( 1 + (0.995 + 0.0980i)T \)
73 \( 1 + (-0.773 - 0.634i)T \)
79 \( 1 + (-0.980 - 0.195i)T \)
83 \( 1 + (0.146 - 0.989i)T \)
89 \( 1 + (-0.956 + 0.290i)T \)
97 \( 1 + (0.923 + 0.382i)T \)
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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

−20.95799253250847937406205783831, −19.965197148909721917215336924102, −19.11617548978511701334519772107, −18.32250539256304870217885959956, −17.79164270186832205369084320146, −17.42072835645740441124657529747, −15.72489062950846905060953623240, −15.65549880877566184987759885133, −14.740105915194475959317843751210, −14.01952420996267140269368925982, −13.18579659577460802364909310855, −12.461278324838257140171678792810, −11.28715997267923150942639015975, −10.92194032072430281326574234353, −10.19420146007467294412816139583, −9.10671000675009032851530958512, −8.37193143117474097246001587629, −7.52382095891424882040338431033, −6.714482092149075499680894975512, −5.81644204337386328586936327417, −5.08039629709769047024976907435, −4.10525890276856532919281612849, −2.79616236272387904715822220350, −2.46629819673067565993216866379, −1.192067679497916349779745118041, 0.77223612099191200581341462495, 1.54425943451967333782638463508, 2.66880705834863013464231431265, 3.88286266227781183340236709721, 4.67526693817168623521104374544, 5.331419779839176097553615662577, 6.24595631787882521801175801567, 7.33343566273411246460181443479, 8.148684060330164264225940461055, 8.72474729254547565624263671016, 9.61236103364370320078256645639, 10.53612103949051895845586163540, 11.28914336894697247970457765974, 11.95701602690601325987823693672, 13.10443333887500441418477740354, 13.53082037074353444004405141351, 14.15044126576825022366251543319, 15.28078440979686890051889647696, 16.08342728730500178107103043561, 16.58445334594497832447807657335, 17.44512252266242781945788312840, 18.1025829736117239197212383465, 18.827122138914288392014369688773, 19.88110367490261697854825004868, 20.559518184704114715304913933144

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