Properties

Label 2-1512-63.4-c1-0-2
Degree $2$
Conductor $1512$
Sign $0.244 - 0.969i$
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  − 3.84·5-s + (0.676 − 2.55i)7-s − 1.80·11-s + (−0.692 + 1.19i)13-s + (0.833 − 1.44i)17-s + (−0.0802 − 0.138i)19-s − 3.20·23-s + 9.75·25-s + (3.78 + 6.54i)29-s + (−1.61 − 2.78i)31-s + (−2.59 + 9.82i)35-s + (1.58 + 2.74i)37-s + (−6.00 + 10.3i)41-s + (3.45 + 5.98i)43-s + (5.71 − 9.90i)47-s + ⋯
L(s)  = 1  − 1.71·5-s + (0.255 − 0.966i)7-s − 0.544·11-s + (−0.192 + 0.332i)13-s + (0.202 − 0.350i)17-s + (−0.0184 − 0.0318i)19-s − 0.667·23-s + 1.95·25-s + (0.701 + 1.21i)29-s + (−0.289 − 0.500i)31-s + (−0.439 + 1.66i)35-s + (0.260 + 0.451i)37-s + (−0.937 + 1.62i)41-s + (0.526 + 0.912i)43-s + (0.834 − 1.44i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.244 - 0.969i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.244 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6739683858\)
\(L(\frac12)\) \(\approx\) \(0.6739683858\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.676 + 2.55i)T \)
good5 \( 1 + 3.84T + 5T^{2} \)
11 \( 1 + 1.80T + 11T^{2} \)
13 \( 1 + (0.692 - 1.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.833 + 1.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0802 + 0.138i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.20T + 23T^{2} \)
29 \( 1 + (-3.78 - 6.54i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.61 + 2.78i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.58 - 2.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.00 - 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.45 - 5.98i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.71 + 9.90i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.37 - 2.38i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.53 - 13.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.60 + 7.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.16 - 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.93T + 71T^{2} \)
73 \( 1 + (6.22 - 10.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.03 - 13.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.45 - 2.51i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.04 + 8.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.18 - 7.25i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.774099919876060307644127127335, −8.485375938183183924285785990382, −8.074351344387114947449993573067, −7.25644470102422816503123750652, −6.78440183707750541572827869214, −5.29351230510296337623504426479, −4.41351514247271504873587247405, −3.83179565167791654958042162428, −2.81171174686899126371292683195, −1.01805339147644963556724093738, 0.33020793646720506917162007636, 2.22604975875602306516597379635, 3.31232403536960456190102133925, 4.17138425397511234140131700582, 5.08139500883012024645142200696, 5.95264240073181500692459276479, 7.09552273442477480745780890735, 7.900507741892754618501777094389, 8.293121260723165368055784633515, 9.095720019470381847665963248790

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