Properties

Label 2-1176-168.149-c0-0-2
Degree $2$
Conductor $1176$
Sign $-0.925 - 0.378i$
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 + (−0.258 + 0.965i)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 + (−0.965 + 0.258i)12-s − 1.41i·13-s + (−0.999 − i)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 + (−0.258 + 0.965i)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 + (−0.965 + 0.258i)12-s − 1.41i·13-s + (−0.999 − i)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

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

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.925 - 0.378i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (1157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ -0.925 - 0.378i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.338785930\)
\(L(\frac12)\) \(\approx\) \(1.338785930\)
\(L(1)\) 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 \( 1 + (0.258 - 0.965i)T \)
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

−10.53892499623865089031631795134, −9.722307749842525116002084424638, −8.422760341999980361273456003908, −7.71723390110773433602622240891, −6.95932560469646087587398797094, −5.91366147022336742530745912288, −5.33359548087387616560037682059, −4.20038622574382325382002568657, −3.34668286858089698264405092974, −2.87495995552795226819473969317, 0.981814426172127204849456122728, 2.07052366309325228295487796832, 3.42876115059211229191791334976, 4.58302899635765665440067393240, 5.11780773223273775934669119598, 6.17560787670356677560915014167, 7.03144935107008582638003360448, 7.79547462097712172457471430239, 8.883376111771230452269644610669, 9.490112048889612040582007586927

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