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Properties

Label 2-264-11.3-c1-0-3
Degree $2$
Conductor $264$
Sign $0.751 + 0.659i$
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 + (1.30 + 0.951i)5-s + (0.690 − 2.12i)7-s + (−0.809 + 0.587i)9-s + (2.80 − 1.76i)11-s + (0.190 − 0.138i)13-s + (0.499 − 1.53i)15-s + (2.11 + 1.53i)17-s + (−1.11 − 3.44i)19-s − 2.23·21-s + 3.47·23-s + (−0.736 − 2.26i)25-s + (0.809 + 0.587i)27-s + (0.618 − 1.90i)29-s + (−2.5 + 1.81i)31-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s + (0.585 + 0.425i)5-s + (0.261 − 0.803i)7-s + (−0.269 + 0.195i)9-s + (0.846 − 0.531i)11-s + (0.0529 − 0.0384i)13-s + (0.129 − 0.397i)15-s + (0.513 + 0.373i)17-s + (−0.256 − 0.789i)19-s − 0.487·21-s + 0.723·23-s + (−0.147 − 0.453i)25-s + (0.155 + 0.113i)27-s + (0.114 − 0.353i)29-s + (−0.449 + 0.326i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.751 + 0.659i$
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: \((2,\ 264,\ (\ :1/2),\ 0.751 + 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27461 - 0.479650i\)
\(L(\frac12)\) \(\approx\) \(1.27461 - 0.479650i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.80 + 1.76i)T \)
good5 \( 1 + (-1.30 - 0.951i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.690 + 2.12i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.190 + 0.138i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.11 - 1.53i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.11 + 3.44i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.47T + 23T^{2} \)
29 \( 1 + (-0.618 + 1.90i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.5 - 1.81i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.07 - 6.37i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.54 - 7.83i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.9T + 43T^{2} \)
47 \( 1 + (-2.28 - 7.02i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (9.35 - 6.79i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.881 - 2.71i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.54 - 2.57i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 14.0T + 67T^{2} \)
71 \( 1 + (-7.16 - 5.20i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.09 + 12.5i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.42 + 1.03i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.42 + 5.39i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 + (-4.78 + 3.47i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.75129053428547926876570532768, −10.95476002851036419846964367513, −10.09837448328689119638345741761, −8.945014666116103270334596193109, −7.83616669167496805942455157985, −6.76758275980106952995241482690, −6.06352948763283613123486280426, −4.60433315582000019413930866466, −3.08501240962548781242511157968, −1.34163356364379587096684064627, 1.85610088045989469098779786115, 3.64198918007368013772081048300, 5.03673494993019992400509103078, 5.76183152306043092707620958607, 7.04150843403177993492083843688, 8.504788972906800848878555437579, 9.272766915912266130439186058178, 9.995263775747147677570193985854, 11.17876726681107133467949827325, 12.06860877175244176979950851600

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