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Properties

Label 2-1176-168.53-c0-0-1
Degree $2$
Conductor $1176$
Sign $-0.378 - 0.925i$
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 + (−0.965 + 0.258i)3-s + (0.499 − 0.866i)4-s + (0.707 + 1.22i)5-s + (0.707 − 0.707i)6-s + 0.999i·8-s + (0.866 − 0.499i)9-s + (−1.22 − 0.707i)10-s + (−0.258 + 0.965i)12-s + 1.41i·13-s + (−1 − 0.999i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)18-s + (1.22 − 0.707i)19-s + 1.41·20-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.965 + 0.258i)3-s + (0.499 − 0.866i)4-s + (0.707 + 1.22i)5-s + (0.707 − 0.707i)6-s + 0.999i·8-s + (0.866 − 0.499i)9-s + (−1.22 − 0.707i)10-s + (−0.258 + 0.965i)12-s + 1.41i·13-s + (−1 − 0.999i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)18-s + (1.22 − 0.707i)19-s + 1.41·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.378 - 0.925i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ -0.378 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5850182093\)
\(L(\frac12)\) \(\approx\) \(0.5850182093\)
\(L(1)\) 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 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 \)
good5 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.06732435428720828756640253756, −9.617176088532957413470881588366, −8.831932477079463479522825277170, −7.32130315856007154420644091577, −6.98377807844384531771129251143, −6.19992450362334696412961950203, −5.52735469781769772655096910686, −4.42395707554104764263749560416, −2.83294550777419548811477524389, −1.54631705780990861602848592824, 0.852383441319613939916401809882, 1.79719917928050684256294939914, 3.31754134115089003049067924610, 4.73571867274999228944556266721, 5.52925115271618508749269518832, 6.29002576914777133318604634515, 7.54001545598417861640102004283, 8.053015105798031298952147086067, 9.095323499424937779058084945253, 9.809947254820757337652285742733

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