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Properties

Label 2-264-11.5-c1-0-3
Degree $2$
Conductor $264$
Sign $0.998 - 0.0475i$
Analytic cond. $2.10805$
Root an. cond. $1.45191$
Motivic weight $1$
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)3-s + (0.190 − 0.587i)5-s + (1.80 − 1.31i)7-s + (0.309 + 0.951i)9-s + (1.69 − 2.85i)11-s + (1.30 + 4.02i)13-s + (0.5 − 0.363i)15-s + (−0.118 + 0.363i)17-s + (1.11 + 0.812i)19-s + 2.23·21-s − 5.47·23-s + (3.73 + 2.71i)25-s + (−0.309 + 0.951i)27-s + (−1.61 + 1.17i)29-s + (−2.5 − 7.69i)31-s + ⋯
L(s)  = 1  + (0.467 + 0.339i)3-s + (0.0854 − 0.262i)5-s + (0.683 − 0.496i)7-s + (0.103 + 0.317i)9-s + (0.509 − 0.860i)11-s + (0.363 + 1.11i)13-s + (0.129 − 0.0937i)15-s + (−0.0286 + 0.0881i)17-s + (0.256 + 0.186i)19-s + 0.487·21-s − 1.14·23-s + (0.747 + 0.542i)25-s + (−0.0594 + 0.183i)27-s + (−0.300 + 0.218i)29-s + (−0.449 − 1.38i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.998 - 0.0475i$
Analytic conductor: \(2.10805\)
Root analytic conductor: \(1.45191\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{264} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 264,\ (\ :1/2),\ 0.998 - 0.0475i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58196 + 0.0376165i\)
\(L(\frac12)\) \(\approx\) \(1.58196 + 0.0376165i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-1.69 + 2.85i)T \)
good5 \( 1 + (-0.190 + 0.587i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.80 + 1.31i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.30 - 4.02i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.118 - 0.363i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.11 - 0.812i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 5.47T + 23T^{2} \)
29 \( 1 + (1.61 - 1.17i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.5 + 7.69i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.42 - 3.94i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.04 + 2.21i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 5.94T + 43T^{2} \)
47 \( 1 + (7.78 + 5.65i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.64 + 8.14i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.11 - 2.26i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.04 - 6.29i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 2.90T + 67T^{2} \)
71 \( 1 + (0.663 - 2.04i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.09 - 5.15i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.92 + 5.93i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.07 - 12.5i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 7.76T + 89T^{2} \)
97 \( 1 + (5.28 + 16.2i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.72436032928882037731733641287, −11.12644095067972738669993913890, −9.999732683709225374666894418485, −8.996044335959091221335905744270, −8.279695761095418457491335652523, −7.13757139669966276848938583356, −5.85696094132759234671971947785, −4.51849568430584481945326550148, −3.57007276742364067301564624494, −1.67813078131380372060368036726, 1.77510488803544109489689803721, 3.18758906129843194180079311809, 4.70091977343419175662852996464, 5.95805558497606906418127805916, 7.14536664806528276416559054461, 8.080907398805565660241410335966, 8.952178158186011862074254171384, 10.04179663258717277850546694854, 11.00620807749204027158944782994, 12.14831447213179960324834968177

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