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Properties

Label 2-1176-1176.989-c0-0-1
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: $\chi_{1176} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ 0.967 - 0.253i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.904902808\)
\(L(\frac12)\) \(\approx\) \(1.904902808\)
\(L(1)\) 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

−9.677308228539332767750085608456, −8.773381583329363817787498820014, −8.343313035810005419020289766339, −7.61840409151411350544663695916, −6.99068535293607896532002428088, −5.43155794105905768701660208822, −4.66809550123662252904540669483, −4.13861748031686010621137143926, −3.26773482731628491530108945789, −1.52378671386282863829721407126, 2.14010540079586024503031326743, 2.68922710630956101413725159302, 3.56747342623340460328252878158, 4.33959264596885262040194215604, 5.77847218492262364167986703892, 6.57147687967328299868833526205, 7.47969132659519591064697312697, 8.190936435389165630910651834622, 9.355126690405173109754772927817, 10.02847989252281458088335024695

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