Properties

Label 2-1512-24.11-c1-0-15
Degree $2$
Conductor $1512$
Sign $-0.975 - 0.218i$
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  + (1.16 + 0.795i)2-s + (0.735 + 1.85i)4-s − 2.09·5-s i·7-s + (−0.618 + 2.75i)8-s + (−2.44 − 1.66i)10-s − 0.961i·11-s + 4.60i·13-s + (0.795 − 1.16i)14-s + (−2.91 + 2.73i)16-s + 3.28i·17-s + 3.66·19-s + (−1.53 − 3.89i)20-s + (0.764 − 1.12i)22-s − 2.40·23-s + ⋯
L(s)  = 1  + (0.826 + 0.562i)2-s + (0.367 + 0.929i)4-s − 0.936·5-s − 0.377i·7-s + (−0.218 + 0.975i)8-s + (−0.774 − 0.526i)10-s − 0.289i·11-s + 1.27i·13-s + (0.212 − 0.312i)14-s + (−0.729 + 0.684i)16-s + 0.795i·17-s + 0.841·19-s + (−0.344 − 0.870i)20-s + (0.162 − 0.239i)22-s − 0.501·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.975 - 0.218i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.975 - 0.218i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.409833901\)
\(L(\frac12)\) \(\approx\) \(1.409833901\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 - 0.795i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 + 2.09T + 5T^{2} \)
11 \( 1 + 0.961iT - 11T^{2} \)
13 \( 1 - 4.60iT - 13T^{2} \)
17 \( 1 - 3.28iT - 17T^{2} \)
19 \( 1 - 3.66T + 19T^{2} \)
23 \( 1 + 2.40T + 23T^{2} \)
29 \( 1 + 9.31T + 29T^{2} \)
31 \( 1 - 1.84iT - 31T^{2} \)
37 \( 1 - 2.29iT - 37T^{2} \)
41 \( 1 + 0.314iT - 41T^{2} \)
43 \( 1 + 8.64T + 43T^{2} \)
47 \( 1 + 2.34T + 47T^{2} \)
53 \( 1 + 7.73T + 53T^{2} \)
59 \( 1 - 7.99iT - 59T^{2} \)
61 \( 1 - 10.6iT - 61T^{2} \)
67 \( 1 - 4.12T + 67T^{2} \)
71 \( 1 - 6.79T + 71T^{2} \)
73 \( 1 - 16.0T + 73T^{2} \)
79 \( 1 - 1.26iT - 79T^{2} \)
83 \( 1 + 2.99iT - 83T^{2} \)
89 \( 1 + 12.8iT - 89T^{2} \)
97 \( 1 - 4.98T + 97T^{2} \)
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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

−9.752063313888223342113890749344, −8.784196319990698871004004021492, −8.003608745494226056561039907935, −7.37570041117983779527760137763, −6.63888998098335324145463370301, −5.76144423667087570173632552082, −4.73979288824627005650425291263, −3.91814903658145108012064555279, −3.39169482534188378731767632522, −1.85603307638737927082819212337, 0.40002091001062807850041873921, 2.00688972714996321857255271776, 3.20683320690640012286943984531, 3.76411485555016258164095784416, 4.96609506644748118573705800558, 5.47799238725662793602541842064, 6.54203755790428005467357384349, 7.54691055567353333805945901933, 8.091402867678996726674247623263, 9.451586296483445461286785416392

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