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Properties

Label 1-264-264.149-r0-0-0
Degree $1$
Conductor $264$
Sign $-0.605 + 0.795i$
Analytic cond. $1.22601$
Root an. cond. $1.22601$
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  + (−0.809 − 0.587i)5-s + (−0.309 + 0.951i)7-s + (−0.809 + 0.587i)13-s + (−0.809 − 0.587i)17-s + (0.309 + 0.951i)19-s − 23-s + (0.309 + 0.951i)25-s + (−0.309 + 0.951i)29-s + (−0.809 + 0.587i)31-s + (0.809 − 0.587i)35-s + (−0.309 + 0.951i)37-s + (0.309 + 0.951i)41-s + 43-s + (−0.309 − 0.951i)47-s + (−0.809 − 0.587i)49-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)5-s + (−0.309 + 0.951i)7-s + (−0.809 + 0.587i)13-s + (−0.809 − 0.587i)17-s + (0.309 + 0.951i)19-s − 23-s + (0.309 + 0.951i)25-s + (−0.309 + 0.951i)29-s + (−0.809 + 0.587i)31-s + (0.809 − 0.587i)35-s + (−0.309 + 0.951i)37-s + (0.309 + 0.951i)41-s + 43-s + (−0.309 − 0.951i)47-s + (−0.809 − 0.587i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(1.22601\)
Root analytic conductor: \(1.22601\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{264} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 264,\ (0:\ ),\ -0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2023201956 + 0.4081049971i\)
\(L(\frac12)\) \(\approx\) \(0.2023201956 + 0.4081049971i\)
\(L(1)\) \(\approx\) \(0.6780405194 + 0.1297981426i\)
\(L(1)\) \(\approx\) \(0.6780405194 + 0.1297981426i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 - T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 - T \)
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

−25.828097913411188228650389760694, −24.26451880470648626390291153305, −23.847253498078399299018361557787, −22.50476042944723481162675654124, −22.33545927173938201918621430556, −20.79626766227555928106242724491, −19.65056529503557184648321041555, −19.5137469753746026445443391702, −18.032121373138983906450605755916, −17.2744877563730168560071255894, −16.09811944947947221699361544763, −15.30090114942282377844877816488, −14.34295400056373797317358678815, −13.30241684869175950264381451758, −12.2844193864403885137514286786, −11.12134266618424135153148854501, −10.44737560962674530217823487248, −9.2857549975204144358094470738, −7.8127732438795286993920272479, −7.25638244710856960640887963641, −6.09418776196175716051576803829, −4.503568703680838857121830837807, −3.65820250296180308755756803076, −2.385738212387472266843589785267, −0.29856266785792178565635919860, 1.824578292371814410826531419284, 3.22174453013562010114623644719, 4.48430331804167238738340193848, 5.464516314479897779283625587896, 6.78440620463908883522734182524, 7.91631282116276392466487490581, 8.90822044173288044783990354496, 9.72227686347032811290031241735, 11.230358553704136839090299894701, 12.11303095835736673322389969875, 12.68334371617260330021577084956, 14.075941068380200859984631007152, 15.09859841562602395919186252549, 16.017944757446690886099201466624, 16.62752348369246314148949953356, 18.00223901531285181992149326148, 18.85317952521474559119995014107, 19.76669167163103018404027934816, 20.51311995244085340946704993571, 21.74906410033805887356098389819, 22.41322611527905999015730348481, 23.53728701860615357192996669226, 24.399860173008280796423329122471, 25.03955957451859259635239460018, 26.2171933864863452441909254328

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