Properties

Degree $2$
Conductor $1512$
Sign $-0.605 - 0.795i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)5-s + (0.5 + 2.59i)7-s − 2·13-s + (−3 + 5.19i)17-s + (0.5 + 0.866i)19-s + (1 + 1.73i)23-s + (0.500 − 0.866i)25-s − 6·29-s + (−0.5 + 0.866i)31-s + (−4 + 3.46i)35-s + (−1 − 1.73i)37-s − 2·41-s + 9·43-s + (−1 − 1.73i)47-s + (−6.5 + 2.59i)49-s + ⋯
L(s)  = 1  + (0.447 + 0.774i)5-s + (0.188 + 0.981i)7-s − 0.554·13-s + (−0.727 + 1.26i)17-s + (0.114 + 0.198i)19-s + (0.208 + 0.361i)23-s + (0.100 − 0.173i)25-s − 1.11·29-s + (−0.0898 + 0.155i)31-s + (−0.676 + 0.585i)35-s + (−0.164 − 0.284i)37-s − 0.312·41-s + 1.37·43-s + (−0.145 − 0.252i)47-s + (−0.928 + 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.605 - 0.795i$
Motivic weight: \(1\)
Character: $\chi_{1512} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.362017775\)
\(L(\frac12)\) \(\approx\) \(1.362017775\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 9T + 43T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (2.5 - 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-9 - 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - T + 97T^{2} \)
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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

−9.644093885762867464957976406461, −9.066760792709629262092133775669, −8.177643758055641779014233342998, −7.33191668927270503132816082471, −6.36613732232801086418716228722, −5.81585193295113263537956030565, −4.85780768877808794439349875568, −3.68952999965831902844520463433, −2.59438019581345098769011813801, −1.82580531957741925980449126311, 0.51286791491399577784254189146, 1.78463812972302615995385256737, 3.04248509884200060114048636225, 4.33671913434408911290457889828, 4.89099901423239278384706201682, 5.79193808526861012654863782863, 6.98992240258271911981076863406, 7.42418892856606446129384482202, 8.486844716697131649978032267011, 9.305199930688179908099916108076

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