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Properties

Label 2-1176-168.131-c0-0-2
Degree $2$
Conductor $1176$
Sign $0.867 + 0.496i$
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 + (−0.608 + 0.793i)3-s + (0.499 + 0.866i)4-s + (0.923 − 0.382i)6-s − 0.999i·8-s + (−0.258 − 0.965i)9-s + (1.22 − 0.707i)11-s + (−0.991 − 0.130i)12-s + (−0.5 + 0.866i)16-s + (−0.923 − 1.60i)17-s + (−0.258 + 0.965i)18-s + (0.662 + 0.382i)19-s − 1.41·22-s + (0.793 + 0.608i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.608 + 0.793i)3-s + (0.499 + 0.866i)4-s + (0.923 − 0.382i)6-s − 0.999i·8-s + (−0.258 − 0.965i)9-s + (1.22 − 0.707i)11-s + (−0.991 − 0.130i)12-s + (−0.5 + 0.866i)16-s + (−0.923 − 1.60i)17-s + (−0.258 + 0.965i)18-s + (0.662 + 0.382i)19-s − 1.41·22-s + (0.793 + 0.608i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.867 + 0.496i$
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: \((2,\ 1176,\ (\ :0),\ 0.867 + 0.496i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5850804200\)
\(L(\frac12)\) \(\approx\) \(0.5850804200\)
\(L(1)\) 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 \( 1 + (0.608 - 0.793i)T \)
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

−9.819942681307502249369518896699, −9.216277136511978260408455901927, −8.699076091128771268855914275810, −7.48335021654121339523569538254, −6.62100402612323655711067868041, −5.81141966957064235441902558586, −4.49729885585474407392313885253, −3.71747633315908806424591599933, −2.62978648986577965650862684006, −0.871160638589655599175500361377, 1.29893822805853055995921851603, 2.21540303311985558193836887070, 4.09638604602968463355976143343, 5.29163302981846879393322157355, 6.17403720684264947808521144057, 6.77297425646585334226091273130, 7.47526663041244491709692743353, 8.318464143276482814715911984141, 9.157617757805327878019575298470, 9.893451523489148475011970118189

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