Properties

Degree $2$
Conductor $1176$
Sign $-0.0725 - 0.997i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (1.70 + 2.95i)5-s + (−0.499 − 0.866i)9-s + (−2.41 + 4.18i)11-s − 1.41·13-s + 3.41·15-s + (−3.12 + 5.40i)17-s + (0.585 + 1.01i)19-s + (0.414 + 0.717i)23-s + (−3.32 + 5.76i)25-s − 0.999·27-s − 8.48·29-s + (5.41 − 9.37i)31-s + (2.41 + 4.18i)33-s + (4.82 + 8.36i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.763 + 1.32i)5-s + (−0.166 − 0.288i)9-s + (−0.727 + 1.26i)11-s − 0.392·13-s + 0.881·15-s + (−0.757 + 1.31i)17-s + (0.134 + 0.232i)19-s + (0.0863 + 0.149i)23-s + (−0.665 + 1.15i)25-s − 0.192·27-s − 1.57·29-s + (0.972 − 1.68i)31-s + (0.420 + 0.727i)33-s + (0.793 + 1.37i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.0725 - 0.997i$
Motivic weight: \(1\)
Character: $\chi_{1176} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -0.0725 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.566478844\)
\(L(\frac12)\) \(\approx\) \(1.566478844\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-1.70 - 2.95i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.41 - 4.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 + (3.12 - 5.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.585 - 1.01i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.414 - 0.717i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.48T + 29T^{2} \)
31 \( 1 + (-5.41 + 9.37i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.82 - 8.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.41T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-0.585 - 1.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.65 - 8.06i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.41 + 9.37i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.94 + 5.10i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.82T + 71T^{2} \)
73 \( 1 + (-1.53 + 2.65i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.82 - 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.31T + 83T^{2} \)
89 \( 1 + (-7.36 - 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−9.869810777972472327672442144579, −9.485654407924237304301040041984, −8.046013770607505518766497382745, −7.56330609880117978091088888913, −6.57033721975889501103001956305, −6.13556913928993743386220624540, −4.89811792055722926384687067144, −3.66163494064543876429030455983, −2.45056394285340542514769624855, −1.94932044323925462013525644150, 0.61045758364407233867175772111, 2.18856373251336111629071287428, 3.25096513587230077695151782078, 4.60382317026410393363965485886, 5.19372654495237789892691975530, 5.88535129589730347306866142881, 7.15734634643921061762521122652, 8.214160366091148721737271308285, 8.911521012704130090289791932401, 9.350194400728402846622538679486

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