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Properties

Label 2-264-11.3-c1-0-5
Degree $2$
Conductor $264$
Sign $-0.944 + 0.329i$
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 + (−2.76 − 2.00i)5-s + (−0.348 + 1.07i)7-s + (−0.809 + 0.587i)9-s + (−0.151 − 3.31i)11-s + (−5.67 + 4.12i)13-s + (−1.05 + 3.24i)15-s + (−3.63 − 2.63i)17-s + (−1.71 − 5.26i)19-s + 1.12·21-s + 6.03·23-s + (2.05 + 6.32i)25-s + (0.809 + 0.587i)27-s + (−0.469 + 1.44i)29-s + (2.66 − 1.93i)31-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s + (−1.23 − 0.897i)5-s + (−0.131 + 0.404i)7-s + (−0.269 + 0.195i)9-s + (−0.0457 − 0.998i)11-s + (−1.57 + 1.14i)13-s + (−0.272 + 0.838i)15-s + (−0.881 − 0.640i)17-s + (−0.392 − 1.20i)19-s + 0.245·21-s + 1.25·23-s + (0.410 + 1.26i)25-s + (0.155 + 0.113i)27-s + (−0.0871 + 0.268i)29-s + (0.478 − 0.347i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $-0.944 + 0.329i$
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: \((2,\ 264,\ (\ :1/2),\ -0.944 + 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0797333 - 0.469868i\)
\(L(\frac12)\) \(\approx\) \(0.0797333 - 0.469868i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.151 + 3.31i)T \)
good5 \( 1 + (2.76 + 2.00i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.348 - 1.07i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (5.67 - 4.12i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.63 + 2.63i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.71 + 5.26i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 6.03T + 23T^{2} \)
29 \( 1 + (0.469 - 1.44i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.66 + 1.93i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.24 + 6.91i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.93 + 9.02i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.96T + 43T^{2} \)
47 \( 1 + (-1.87 - 5.78i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.952 - 0.691i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.06 + 3.29i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.42 - 1.76i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 + (13.1 + 9.54i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.29 - 3.98i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.87 + 1.36i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.459 - 0.333i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 13.9T + 89T^{2} \)
97 \( 1 + (-8.17 + 5.93i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.62975592598615499965103107832, −11.03953081161000638035987508965, −9.170448282484274393041096471068, −8.783387063460769551611721500271, −7.51790336135722744778373307783, −6.78445880538031367317480515356, −5.20027865462852633737369343168, −4.33908717039591673386875579846, −2.59860614528398558229479688816, −0.36231611637422432710624801560, 2.82349249752504298581670794261, 4.01032514935605213723537676362, 4.98440240005803145221169600393, 6.64763859245716155185655388474, 7.46011933562942251298260687834, 8.338155550644496841861408882862, 9.976083149018410119509828366338, 10.35189986051393527954822489970, 11.38883166133916155354009253744, 12.23033264201180469092895087394

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