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Properties

Label 2-1452-11.4-c1-0-5
Degree $2$
Conductor $1452$
Sign $0.642 - 0.766i$
Analytic cond. $11.5942$
Root an. cond. $3.40503$
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 + (−2.30 + 1.67i)5-s + (−1.30 − 4.02i)7-s + (−0.809 − 0.587i)9-s + (−1.42 − 1.03i)13-s + (−0.881 − 2.71i)15-s + (3.73 − 2.71i)17-s + (−1.88 + 5.79i)19-s + 4.23·21-s + 4.23·23-s + (0.972 − 2.99i)25-s + (0.809 − 0.587i)27-s + (1.38 + 4.25i)29-s + (6.97 + 5.06i)31-s + (9.78 + 7.10i)35-s + ⋯
L(s)  = 1  + (−0.178 + 0.549i)3-s + (−1.03 + 0.750i)5-s + (−0.494 − 1.52i)7-s + (−0.269 − 0.195i)9-s + (−0.395 − 0.287i)13-s + (−0.227 − 0.700i)15-s + (0.906 − 0.658i)17-s + (−0.431 + 1.32i)19-s + 0.924·21-s + 0.883·23-s + (0.194 − 0.598i)25-s + (0.155 − 0.113i)27-s + (0.256 + 0.789i)29-s + (1.25 + 0.909i)31-s + (1.65 + 1.20i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(11.5942\)
Root analytic conductor: \(3.40503\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1452} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.026001368\)
\(L(\frac12)\) \(\approx\) \(1.026001368\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (2.30 - 1.67i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.30 + 4.02i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.42 + 1.03i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.73 + 2.71i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.88 - 5.79i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.23T + 23T^{2} \)
29 \( 1 + (-1.38 - 4.25i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-6.97 - 5.06i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.54 + 7.83i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.163 - 0.502i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.527T + 43T^{2} \)
47 \( 1 + (-0.427 + 1.31i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-10.9 - 7.97i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.73 - 8.42i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.309 - 0.224i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 6.85T + 67T^{2} \)
71 \( 1 + (-2.92 + 2.12i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.381 - 1.17i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.89 - 5.73i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.28 + 3.83i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 + (4.92 + 3.57i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.961722698051705107554708857412, −8.849262813681695432186508489994, −7.76011891015995818102922287126, −7.29475766572205055263331669229, −6.59111010289174176298953117800, −5.39153649692368389159308569067, −4.33426423396462684453909511639, −3.62717983498742547942509761642, −2.99967479143601670779681346381, −0.849917056081675362121312011469, 0.61644949194073565150752875739, 2.22719406730250593612383590840, 3.18161751494073813738009851123, 4.48004347760732936880125161961, 5.23038417203911084280758451784, 6.17042326380821752447031026226, 6.93052664226883338009603833378, 8.036693203450819884871617767707, 8.473060060565275133297339175162, 9.210525784426922043898146515122

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