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Properties

Label 2-264-264.227-c0-0-1
Degree $2$
Conductor $264$
Sign $0.970 - 0.242i$
Analytic cond. $0.131753$
Root an. cond. $0.362978$
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)2-s + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)6-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.309 − 0.951i)11-s + (0.809 − 0.587i)12-s + (−0.809 + 0.587i)16-s + (−0.5 + 0.363i)17-s − 0.999·18-s + (−1.80 − 0.587i)19-s + (0.809 − 0.587i)22-s + 24-s + (−0.309 + 0.951i)25-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)6-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.309 − 0.951i)11-s + (0.809 − 0.587i)12-s + (−0.809 + 0.587i)16-s + (−0.5 + 0.363i)17-s − 0.999·18-s + (−1.80 − 0.587i)19-s + (0.809 − 0.587i)22-s + 24-s + (−0.309 + 0.951i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(0.131753\)
Root analytic conductor: \(0.362978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{264} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 264,\ (\ :0),\ 0.970 - 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9878336230\)
\(L(\frac12)\) \(\approx\) \(0.9878336230\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.309 + 0.951i)T \)
good5 \( 1 + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - 1.17iT - T^{2} \)
47 \( 1 + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 - 1.17iT - T^{2} \)
97 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.53754645894466774899537708340, −11.42029206930293426009731281449, −10.88320607462122253728744546217, −8.863944082603331619986402463646, −8.183784225832969237458163601321, −6.99820737920006169748928740343, −6.28867917791229953829435959466, −5.34806349899617237912977398325, −3.90665421499112695506344499848, −2.34333803514186724896722920634, 2.37362098857690372097412503603, 3.99531835212255375205442183375, 4.60400992516194197325476296894, 5.86363959105077971020893479102, 6.79997513346974898580798213337, 8.598551555791422644854415597380, 9.715693376172804525377898265638, 10.36469697255195961855380329691, 11.23219503082205516994490223784, 12.11760008892500796427656830928

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