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Properties

Label 2-1512-7.4-c1-0-3
Degree $2$
Conductor $1512$
Sign $-0.421 - 0.906i$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
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  + (−1.23 − 2.13i)5-s + (−0.0665 + 2.64i)7-s + (−0.433 + 0.750i)11-s − 1.46·13-s + (0.933 − 1.61i)17-s + (1.29 + 2.24i)19-s + (−0.796 − 1.38i)23-s + (−0.527 + 0.912i)25-s − 5.78·29-s + (−4.78 + 8.28i)31-s + (5.71 − 3.11i)35-s + (1.66 + 2.87i)37-s − 2.78·41-s − 10.7·43-s + (6.74 + 11.6i)47-s + ⋯
L(s)  = 1  + (−0.550 − 0.952i)5-s + (−0.0251 + 0.999i)7-s + (−0.130 + 0.226i)11-s − 0.405·13-s + (0.226 − 0.392i)17-s + (0.297 + 0.515i)19-s + (−0.166 − 0.287i)23-s + (−0.105 + 0.182i)25-s − 1.07·29-s + (−0.859 + 1.48i)31-s + (0.966 − 0.526i)35-s + (0.272 + 0.472i)37-s − 0.434·41-s − 1.63·43-s + (0.983 + 1.70i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.421 - 0.906i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.421 - 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6804903940\)
\(L(\frac12)\) \(\approx\) \(0.6804903940\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (0.0665 - 2.64i)T \)
good5 \( 1 + (1.23 + 2.13i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.433 - 0.750i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.46T + 13T^{2} \)
17 \( 1 + (-0.933 + 1.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.29 - 2.24i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.796 + 1.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.78T + 29T^{2} \)
31 \( 1 + (4.78 - 8.28i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.66 - 2.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.78T + 41T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 + (-6.74 - 11.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.32 - 5.75i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.21 + 5.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.09 - 5.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.850 - 1.47i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.14T + 71T^{2} \)
73 \( 1 + (5.08 - 8.80i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.39 - 5.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.75T + 83T^{2} \)
89 \( 1 + (-4.96 - 8.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.81T + 97T^{2} \)
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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

−9.514769363229633549306872524613, −8.933314178038921164325983106971, −8.198756855277212268041469845200, −7.50212161332186545461458230608, −6.44856562437583761155957291190, −5.36432028883190822890385737687, −4.92753828317659976760270681000, −3.80917119153236613241465311024, −2.70020597148036564409983213027, −1.43456190479989605750685885457, 0.26910885924101985247619384120, 2.01587897436809391068645474809, 3.36216089768852145005543360589, 3.83983239809628203155717075112, 5.01079671392943101702314046088, 6.04318194664160977199137965734, 7.08827344318502441371901167377, 7.40032177278465667505361775869, 8.225239510806663814083585695342, 9.356125490138188125009874602330

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