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Properties

Label 2-1176-168.131-c0-0-5
Degree $2$
Conductor $1176$
Sign $0.999 - 0.0350i$
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 + (−0.130 − 0.991i)3-s + (0.499 + 0.866i)4-s + (0.382 − 0.923i)6-s + 0.999i·8-s + (−0.965 + 0.258i)9-s + (1.22 − 0.707i)11-s + (0.793 − 0.608i)12-s + (−0.5 + 0.866i)16-s + (−0.382 − 0.662i)17-s + (−0.965 − 0.258i)18-s + (1.60 + 0.923i)19-s + 1.41·22-s + (0.991 − 0.130i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.130 − 0.991i)3-s + (0.499 + 0.866i)4-s + (0.382 − 0.923i)6-s + 0.999i·8-s + (−0.965 + 0.258i)9-s + (1.22 − 0.707i)11-s + (0.793 − 0.608i)12-s + (−0.5 + 0.866i)16-s + (−0.382 − 0.662i)17-s + (−0.965 − 0.258i)18-s + (1.60 + 0.923i)19-s + 1.41·22-s + (0.991 − 0.130i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.999 - 0.0350i$
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: \((2,\ 1176,\ (\ :0),\ 0.999 - 0.0350i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.686193029\)
\(L(\frac12)\) \(\approx\) \(1.686193029\)
\(L(1)\) 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 \( 1 + (0.130 + 0.991i)T \)
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 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + 1.84T + T^{2} \)
43 \( 1 + 1.41T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + 0.765T + T^{2} \)
89 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + 1.84iT - T^{2} \)
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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

−9.954082398750587821962660506114, −8.782325788824092655306664196915, −8.163480685754652888161463359432, −7.24969878495946062107565334848, −6.61811664447869132427178126872, −5.87228854081291024486155960113, −5.08587995431325615091189320310, −3.78982328642598243145098954673, −2.93114561150627256665044767706, −1.55386461065253944314109226859, 1.65016294515927629137116902237, 3.12988592206788613255999555030, 3.83370808906513824137236758346, 4.74290011258802160675799765390, 5.40884084680106131933430881579, 6.43933980993900553381096554542, 7.19012856129294059543652793913, 8.649478948575938813922899370595, 9.542961087132898881569454332658, 9.911870825035617164522026569326

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