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Properties

Label 1-1053-1053.254-r0-0-0
Degree $1$
Conductor $1053$
Sign $0.941 - 0.336i$
Analytic cond. $4.89011$
Root an. cond. $4.89011$
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  + (0.549 + 0.835i)2-s + (−0.396 + 0.918i)4-s + (0.230 − 0.973i)5-s + (0.918 − 0.396i)7-s + (−0.984 + 0.173i)8-s + (0.939 − 0.342i)10-s + (−0.957 − 0.286i)11-s + (0.835 + 0.549i)14-s + (−0.686 − 0.727i)16-s + (0.173 − 0.984i)17-s + (0.642 + 0.766i)19-s + (0.802 + 0.597i)20-s + (−0.286 − 0.957i)22-s + (0.396 − 0.918i)23-s + (−0.893 − 0.448i)25-s + ⋯
L(s)  = 1  + (0.549 + 0.835i)2-s + (−0.396 + 0.918i)4-s + (0.230 − 0.973i)5-s + (0.918 − 0.396i)7-s + (−0.984 + 0.173i)8-s + (0.939 − 0.342i)10-s + (−0.957 − 0.286i)11-s + (0.835 + 0.549i)14-s + (−0.686 − 0.727i)16-s + (0.173 − 0.984i)17-s + (0.642 + 0.766i)19-s + (0.802 + 0.597i)20-s + (−0.286 − 0.957i)22-s + (0.396 − 0.918i)23-s + (−0.893 − 0.448i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $0.941 - 0.336i$
Analytic conductor: \(4.89011\)
Root analytic conductor: \(4.89011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1053,\ (0:\ ),\ 0.941 - 0.336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.817023816 - 0.3144007408i\)
\(L(\frac12)\) \(\approx\) \(1.817023816 - 0.3144007408i\)
\(L(1)\) \(\approx\) \(1.365051569 + 0.2024970932i\)
\(L(1)\) \(\approx\) \(1.365051569 + 0.2024970932i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.549 + 0.835i)T \)
5 \( 1 + (0.230 - 0.973i)T \)
7 \( 1 + (0.918 - 0.396i)T \)
11 \( 1 + (-0.957 - 0.286i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (0.642 + 0.766i)T \)
23 \( 1 + (0.396 - 0.918i)T \)
29 \( 1 + (-0.893 - 0.448i)T \)
31 \( 1 + (0.918 + 0.396i)T \)
37 \( 1 + (-0.984 - 0.173i)T \)
41 \( 1 + (-0.448 - 0.893i)T \)
43 \( 1 + (0.686 + 0.727i)T \)
47 \( 1 + (-0.918 + 0.396i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.957 - 0.286i)T \)
61 \( 1 + (0.597 - 0.802i)T \)
67 \( 1 + (-0.998 - 0.0581i)T \)
71 \( 1 + (0.642 - 0.766i)T \)
73 \( 1 + (0.984 - 0.173i)T \)
79 \( 1 + (-0.0581 - 0.998i)T \)
83 \( 1 + (-0.998 + 0.0581i)T \)
89 \( 1 + (0.642 + 0.766i)T \)
97 \( 1 + (-0.957 - 0.286i)T \)
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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

−21.38313896675835988796564997511, −21.11154548829858491328146928219, −20.104497496110893474200914069037, −19.22237062528493282025575061104, −18.52559066779554460103758038338, −17.93172366942419297399074060274, −17.25143215217123843407753767744, −15.54110989478388949904898810062, −15.19476218524685862448390974697, −14.429499866282489779792622156282, −13.60285407508977411761020203125, −12.96041515137011584448822009467, −11.840923043071885087262889827472, −11.277275212116937005869715072947, −10.52929954037596072010068019900, −9.87888622520041986684189575316, −8.83555645843387156339520268922, −7.78320307208126147608441026199, −6.82657210591921510295316410856, −5.63362583409676924257292118845, −5.18981274655530821249648348757, −4.02599607488342944413468068111, −3.03082166873128360795023574852, −2.28566646039050097527161753740, −1.39893673031553747860494253212, 0.65134167692708109641222736623, 2.08985910969991543609676314794, 3.32176158970261588496679688525, 4.425582171666970506060835134962, 5.12009210399666630649023208351, 5.60163976122166015227337506786, 6.84826590911257808423744114141, 7.83294296689704830500171282606, 8.251390875231588728144720493570, 9.18247743931387589895234583267, 10.18840194727220528087748905994, 11.37422831525479675760804629298, 12.16115596765537650344209484196, 12.959923559653079935352372627473, 13.751980852191997604475873875819, 14.23936572455387201605356069907, 15.26060140593619767348911591176, 16.14530978620839899958989293311, 16.55714201001315923542520197607, 17.49940121145126855131111951028, 18.03708393442989287992542958851, 18.96640186377930219953829277350, 20.445733169707989846936409734059, 20.86126461300931164894360575586, 21.22996606435128188594652609099

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