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Properties

Label 2-1176-168.59-c0-0-7
Degree $2$
Conductor $1176$
Sign $0.664 + 0.747i$
Analytic cond. $0.586900$
Root an. cond. $0.766094$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.991 + 0.130i)3-s + (0.499 − 0.866i)4-s + (0.923 − 0.382i)6-s − 0.999i·8-s + (0.965 + 0.258i)9-s + (−1.22 − 0.707i)11-s + (0.608 − 0.793i)12-s + (−0.5 − 0.866i)16-s + (−0.923 + 1.60i)17-s + (0.965 − 0.258i)18-s + (−0.662 + 0.382i)19-s − 1.41·22-s + (0.130 − 0.991i)24-s + (−0.5 + 0.866i)25-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.991 + 0.130i)3-s + (0.499 − 0.866i)4-s + (0.923 − 0.382i)6-s − 0.999i·8-s + (0.965 + 0.258i)9-s + (−1.22 − 0.707i)11-s + (0.608 − 0.793i)12-s + (−0.5 − 0.866i)16-s + (−0.923 + 1.60i)17-s + (0.965 − 0.258i)18-s + (−0.662 + 0.382i)19-s − 1.41·22-s + (0.130 − 0.991i)24-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.664 + 0.747i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ 0.664 + 0.747i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.177450315\)
\(L(\frac12)\) \(\approx\) \(2.177450315\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.991 - 0.130i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - 0.765T + T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.60 + 0.923i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + 1.84T + T^{2} \)
89 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + 0.765iT - T^{2} \)
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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

−10.07940954454811518370876304604, −9.069536942660358507897657330812, −8.268990322634649869411046069351, −7.45979121386736352584169872112, −6.33197752842882123838850477479, −5.52439772870617366434208850944, −4.38513234254987449020773655164, −3.68450258381241490873032178124, −2.68036814911113335927034453918, −1.79123172966811295921957262070, 2.36531938906347591253055447549, 2.72852581456127403536610247468, 4.19420749039456483971863135388, 4.71241293402517279402986139957, 5.84908045460045185878652982551, 7.04731016343117792291615220093, 7.38012488776992080600317370556, 8.301775646923199554949197382445, 9.033903324005790650506444041439, 9.998409615839924992816174008060

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