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Properties

Label 2-264-11.3-c1-0-1
Degree $2$
Conductor $264$
Sign $0.834 - 0.551i$
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  + (−0.309 − 0.951i)3-s + (1.95 + 1.41i)5-s + (−1.46 + 4.49i)7-s + (−0.809 + 0.587i)9-s + (0.960 + 3.17i)11-s + (4.86 − 3.53i)13-s + (0.745 − 2.29i)15-s + (−0.720 − 0.523i)17-s + (−1.02 − 3.15i)19-s + 4.72·21-s + 0.204·23-s + (0.254 + 0.782i)25-s + (0.809 + 0.587i)27-s + (−2.95 + 9.10i)29-s + (7.37 − 5.36i)31-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s + (0.873 + 0.634i)5-s + (−0.552 + 1.69i)7-s + (−0.269 + 0.195i)9-s + (0.289 + 0.957i)11-s + (1.34 − 0.980i)13-s + (0.192 − 0.592i)15-s + (−0.174 − 0.126i)17-s + (−0.234 − 0.723i)19-s + 1.03·21-s + 0.0427·23-s + (0.0508 + 0.156i)25-s + (0.155 + 0.113i)27-s + (−0.549 + 1.69i)29-s + (1.32 − 0.962i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.834 - 0.551i$
Analytic conductor: \(2.10805\)
Root analytic conductor: \(1.45191\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{264} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 264,\ (\ :1/2),\ 0.834 - 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26015 + 0.379003i\)
\(L(\frac12)\) \(\approx\) \(1.26015 + 0.379003i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.960 - 3.17i)T \)
good5 \( 1 + (-1.95 - 1.41i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.46 - 4.49i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.86 + 3.53i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.720 + 0.523i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.02 + 3.15i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.204T + 23T^{2} \)
29 \( 1 + (2.95 - 9.10i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-7.37 + 5.36i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.445 + 1.37i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.09 - 3.36i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.79T + 43T^{2} \)
47 \( 1 + (2.83 + 8.72i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.76 + 2.73i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.305 - 0.940i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.20 + 3.78i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 6.17T + 67T^{2} \)
71 \( 1 + (5.50 + 3.99i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.19 + 3.67i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.67 + 2.66i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.08 - 5.87i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 0.930T + 89T^{2} \)
97 \( 1 + (2.36 - 1.71i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.15439089072707104730805145637, −11.18402160055838755728118437479, −10.10839284080270267549274784488, −9.178004756243803460011744186038, −8.294384149291116767885087141292, −6.75721603422762035784350102558, −6.14961921388431389755952977069, −5.23119739920303061372567060326, −3.08455089433770825477294176015, −2.02341007596830963999430240751, 1.21070720301408581523079005809, 3.58966176002755736202406950623, 4.40323751386956193417948310426, 5.94741203513246548160926754942, 6.59468307457270616492348538352, 8.166933611088299108696717854361, 9.168012433168044713346769691606, 10.01025276814548309608982604560, 10.78085517671165112252528339677, 11.67263445348175787346434254387

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