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Properties

Label 2-1176-1176.893-c0-0-0
Degree $2$
Conductor $1176$
Sign $0.967 + 0.253i$
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  + (−0.733 + 0.680i)2-s + (−0.988 + 0.149i)3-s + (0.0747 − 0.997i)4-s + (0.698 − 1.77i)5-s + (0.623 − 0.781i)6-s + (0.365 + 0.930i)7-s + (0.623 + 0.781i)8-s + (0.955 − 0.294i)9-s + (0.698 + 1.77i)10-s + (0.142 + 0.0440i)11-s + (0.0747 + 0.997i)12-s + (−0.900 − 0.433i)14-s + (−0.425 + 1.86i)15-s + (−0.988 − 0.149i)16-s + (−0.5 + 0.866i)18-s + ⋯
L(s)  = 1  + (−0.733 + 0.680i)2-s + (−0.988 + 0.149i)3-s + (0.0747 − 0.997i)4-s + (0.698 − 1.77i)5-s + (0.623 − 0.781i)6-s + (0.365 + 0.930i)7-s + (0.623 + 0.781i)8-s + (0.955 − 0.294i)9-s + (0.698 + 1.77i)10-s + (0.142 + 0.0440i)11-s + (0.0747 + 0.997i)12-s + (−0.900 − 0.433i)14-s + (−0.425 + 1.86i)15-s + (−0.988 − 0.149i)16-s + (−0.5 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.967 + 0.253i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (893, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ 0.967 + 0.253i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6426425205\)
\(L(\frac12)\) \(\approx\) \(0.6426425205\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.733 - 0.680i)T \)
3 \( 1 + (0.988 - 0.149i)T \)
7 \( 1 + (-0.365 - 0.930i)T \)
good5 \( 1 + (-0.698 + 1.77i)T + (-0.733 - 0.680i)T^{2} \)
11 \( 1 + (-0.142 - 0.0440i)T + (0.826 + 0.563i)T^{2} \)
13 \( 1 + (0.900 - 0.433i)T^{2} \)
17 \( 1 + (-0.365 + 0.930i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.365 - 0.930i)T^{2} \)
29 \( 1 + (-1.32 - 0.636i)T + (0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.988 - 0.149i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.0747 + 0.997i)T^{2} \)
53 \( 1 + (0.0747 - 0.997i)T + (-0.988 - 0.149i)T^{2} \)
59 \( 1 + (0.365 + 0.930i)T + (-0.733 + 0.680i)T^{2} \)
61 \( 1 + (0.988 - 0.149i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (0.914 + 0.848i)T + (0.0747 + 0.997i)T^{2} \)
79 \( 1 + (0.365 + 0.632i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.326 + 1.42i)T + (-0.900 - 0.433i)T^{2} \)
89 \( 1 + (-0.826 + 0.563i)T^{2} \)
97 \( 1 - 1.65T + T^{2} \)
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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

−9.717272851332642124229796014600, −9.103348154214040484202823449381, −8.478821876816497115633952239409, −7.62258258466886153842575907801, −6.17516282986141900534582417086, −5.94695162597851021004142559671, −4.88932403142334701347267735335, −4.62613076126677661470282982205, −2.02038608907156560305303189554, −0.972851500848864548771385621624, 1.36090121894663116918278094264, 2.58379341843913479807886318643, 3.63873586621911270825863868933, 4.75399134421790808208828859591, 6.16822727697681151650607335798, 6.84281279240343809390238268839, 7.31201047150682318607648614470, 8.312865073559473991263288319817, 9.715566113828704395988226979201, 10.31645456676235871787570710224

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