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Properties

Label 2-1176-168.131-c0-0-0
Degree $2$
Conductor $1176$
Sign $-0.262 - 0.964i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.262 - 0.964i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (803, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ -0.262 - 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.188538625\)
\(L(\frac12)\) \(\approx\) \(1.188538625\)
\(L(1)\) 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

−10.44321183214863070980259726245, −9.607214037068532328040371978511, −8.087295509996374724895327105692, −7.69518396032347023160447026441, −6.64327798894461061444385415327, −5.85964711990696560736399128705, −5.24050188249656756114191382248, −4.37516632514252910901341631463, −3.42999765719725727154454520528, −1.95343433030854037321134442973, 0.936286070773726277413058010821, 2.53803290672045879997829642185, 3.52243861200202814211775070698, 4.85336860502133195317263581352, 5.34948952789893056497577598906, 6.01200842452673004141420232282, 7.17130015459564285876499415443, 7.68810338474980785421293452542, 9.290365448907137855867605304152, 10.01059616246745940303261948100

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