Properties

Degree $2$
Conductor $1512$
Sign $0.386 - 0.922i$
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 + (2 − 1.73i)7-s + (−2 + 3.46i)11-s − 13-s + (2 + 3.46i)19-s + (3 + 5.19i)23-s + (0.500 − 0.866i)25-s + (−2.5 + 4.33i)31-s + (5 + 1.73i)35-s + (−0.5 − 0.866i)37-s + 4·41-s − 43-s + (2 + 3.46i)47-s + (1.00 − 6.92i)49-s + (−3 + 5.19i)53-s + ⋯
L(s)  = 1  + (0.447 + 0.774i)5-s + (0.755 − 0.654i)7-s + (−0.603 + 1.04i)11-s − 0.277·13-s + (0.458 + 0.794i)19-s + (0.625 + 1.08i)23-s + (0.100 − 0.173i)25-s + (−0.449 + 0.777i)31-s + (0.845 + 0.292i)35-s + (−0.0821 − 0.142i)37-s + 0.624·41-s − 0.152·43-s + (0.291 + 0.505i)47-s + (0.142 − 0.989i)49-s + (−0.412 + 0.713i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.386 - 0.922i$
Motivic weight: \(1\)
Character: $\chi_{1512} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.823501840\)
\(L(\frac12)\) \(\approx\) \(1.823501840\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (-2 - 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9T + 97T^{2} \)
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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

−9.823348706204214982254806881826, −8.892714363348812050889419324501, −7.60788226061128595362499722709, −7.48310748097923436102757240735, −6.47515135539313978584301072896, −5.41376898939079185091165290225, −4.69136893603325239482987738266, −3.59649481363211597308414624373, −2.48534898001024316624520542329, −1.44461821182599522797268036039, 0.75772461077159913566858037421, 2.11591438405784893882059971325, 3.07720860368886879697045209395, 4.51214234160735207134550349650, 5.24800358713264851021598547840, 5.77267080365393015011603722200, 6.90126094104091601505541024691, 7.947966547760633412314276520459, 8.602451448696083889421067042262, 9.142562256287727828682841364170

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