Properties

Degree $2$
Conductor $1512$
Sign $-0.968 + 0.250i$
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 + 3·13-s + (−2 − 3.46i)17-s + (−2 + 3.46i)19-s + (−1 + 1.73i)23-s + (0.500 + 0.866i)25-s − 4·29-s + (−0.5 − 0.866i)31-s + (−0.999 − 5.19i)35-s + (1.5 − 2.59i)37-s − 8·41-s − 43-s + (−4 + 6.92i)47-s + (1.00 − 6.92i)49-s + ⋯
L(s)  = 1  + (−0.447 + 0.774i)5-s + (−0.755 + 0.654i)7-s + 0.832·13-s + (−0.485 − 0.840i)17-s + (−0.458 + 0.794i)19-s + (−0.208 + 0.361i)23-s + (0.100 + 0.173i)25-s − 0.742·29-s + (−0.0898 − 0.155i)31-s + (−0.169 − 0.878i)35-s + (0.246 − 0.427i)37-s − 1.24·41-s − 0.152·43-s + (−0.583 + 1.01i)47-s + (0.142 − 0.989i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.968 + 0.250i$
Motivic weight: \(1\)
Character: $\chi_{1512} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2860300113\)
\(L(\frac12)\) \(\approx\) \(0.2860300113\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7 + 12.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \)
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

−9.815920094976088700239137358167, −9.172973259470855292700232571058, −8.305514063770918245731476702495, −7.46415474756975541248291397252, −6.58158678229828667016807997642, −6.03212921867064587229323274154, −4.96540671667323263208021363650, −3.68115449615579019550638016649, −3.14077846542599206315126721488, −1.91087569723324215504985053479, 0.11188057592845452424374525565, 1.47100699858320982647976328624, 3.02093833265272668689587033858, 4.04341542234415420150493887393, 4.60124758653951207869590886746, 5.86218632339592811783623248602, 6.59390270747370262162111245295, 7.41849947930499806461184205500, 8.495147465316826516108391790555, 8.798819876060741563852154733413

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