Properties

Degree $2$
Conductor $1512$
Sign $0.840 + 0.541i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 + 0.633i)5-s + (−2.62 − 0.295i)7-s + (1.55 + 2.69i)11-s − 2.32·13-s + (−3.09 − 5.36i)17-s + (3.36 − 5.82i)19-s + (2.33 − 4.04i)23-s + (2.23 + 3.86i)25-s + 9.24·29-s + (−0.326 − 0.565i)31-s + (1.14 − 1.55i)35-s + (−5.41 + 9.38i)37-s + 12.4·41-s + 1.77·43-s + (5.50 − 9.53i)47-s + ⋯
L(s)  = 1  + (−0.163 + 0.283i)5-s + (−0.993 − 0.111i)7-s + (0.469 + 0.813i)11-s − 0.644·13-s + (−0.751 − 1.30i)17-s + (0.771 − 1.33i)19-s + (0.486 − 0.842i)23-s + (0.446 + 0.773i)25-s + 1.71·29-s + (−0.0586 − 0.101i)31-s + (0.194 − 0.263i)35-s + (−0.891 + 1.54i)37-s + 1.94·41-s + 0.270·43-s + (0.803 − 1.39i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.840 + 0.541i$
Motivic weight: \(1\)
Character: $\chi_{1512} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.840 + 0.541i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.320969945\)
\(L(\frac12)\) \(\approx\) \(1.320969945\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 + 0.295i)T \)
good5 \( 1 + (0.366 - 0.633i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.55 - 2.69i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.32T + 13T^{2} \)
17 \( 1 + (3.09 + 5.36i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.36 + 5.82i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.33 + 4.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 9.24T + 29T^{2} \)
31 \( 1 + (0.326 + 0.565i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.41 - 9.38i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 12.4T + 41T^{2} \)
43 \( 1 - 1.77T + 43T^{2} \)
47 \( 1 + (-5.50 + 9.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.398 - 0.689i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.290 - 0.503i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.264 + 0.457i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.59 + 9.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.34T + 71T^{2} \)
73 \( 1 + (1.48 + 2.57i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.68 + 8.11i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.0T + 83T^{2} \)
89 \( 1 + (-5.89 + 10.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.77T + 97T^{2} \)
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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

−9.266004599932750964302628346913, −8.960570561378890704721026581358, −7.49349495090752061175070197730, −6.95530184654912952068200655373, −6.46554426658708026537751133168, −5.03199987043012171155701248155, −4.49377274589484376999374986843, −3.12404453591232983900965592616, −2.52410306231776310237805422442, −0.65922131214681156877846534642, 1.01615110955226368189611019865, 2.55742229388302588839466958290, 3.58420143966260572218206430463, 4.33738387554476886678627824894, 5.65861514360589766329905655095, 6.16903590059976336896897369616, 7.10108154475246072347938405024, 8.015860481313747266421973747037, 8.843656513978856803722508227766, 9.426387211093172001019400327728

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