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Properties

Label 2-1176-1176.53-c0-0-1
Degree $2$
Conductor $1176$
Sign $0.775 - 0.631i$
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  + (0.826 + 0.563i)2-s + (−0.733 − 0.680i)3-s + (0.365 + 0.930i)4-s + (0.142 − 0.0440i)5-s + (−0.222 − 0.974i)6-s + (0.955 + 0.294i)7-s + (−0.222 + 0.974i)8-s + (0.0747 + 0.997i)9-s + (0.142 + 0.0440i)10-s + (0.0546 − 0.728i)11-s + (0.365 − 0.930i)12-s + (0.623 + 0.781i)14-s + (−0.134 − 0.0648i)15-s + (−0.733 + 0.680i)16-s + (−0.499 + 0.866i)18-s + ⋯
L(s)  = 1  + (0.826 + 0.563i)2-s + (−0.733 − 0.680i)3-s + (0.365 + 0.930i)4-s + (0.142 − 0.0440i)5-s + (−0.222 − 0.974i)6-s + (0.955 + 0.294i)7-s + (−0.222 + 0.974i)8-s + (0.0747 + 0.997i)9-s + (0.142 + 0.0440i)10-s + (0.0546 − 0.728i)11-s + (0.365 − 0.930i)12-s + (0.623 + 0.781i)14-s + (−0.134 − 0.0648i)15-s + (−0.733 + 0.680i)16-s + (−0.499 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.775 - 0.631i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ 0.775 - 0.631i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.471144028\)
\(L(\frac12)\) \(\approx\) \(1.471144028\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.826 - 0.563i)T \)
3 \( 1 + (0.733 + 0.680i)T \)
7 \( 1 + (-0.955 - 0.294i)T \)
good5 \( 1 + (-0.142 + 0.0440i)T + (0.826 - 0.563i)T^{2} \)
11 \( 1 + (-0.0546 + 0.728i)T + (-0.988 - 0.149i)T^{2} \)
13 \( 1 + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + (-0.955 + 0.294i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.955 - 0.294i)T^{2} \)
29 \( 1 + (-1.03 - 1.29i)T + (-0.222 + 0.974i)T^{2} \)
31 \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.733 + 0.680i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (-0.365 - 0.930i)T^{2} \)
53 \( 1 + (0.365 + 0.930i)T + (-0.733 + 0.680i)T^{2} \)
59 \( 1 + (0.955 + 0.294i)T + (0.826 + 0.563i)T^{2} \)
61 \( 1 + (0.733 + 0.680i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (0.367 - 0.250i)T + (0.365 - 0.930i)T^{2} \)
79 \( 1 + (0.955 + 1.65i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (1.48 + 0.716i)T + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (0.988 - 0.149i)T^{2} \)
97 \( 1 + 1.97T + T^{2} \)
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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

−10.32699301252974091620242833001, −8.881995354682936154858930835526, −8.122376834913407658295974294429, −7.50596694288350777997875093444, −6.54028244456563011568735635576, −5.81597999041366004614471887865, −5.15291364539000128079544470867, −4.30385710799786558744813763944, −2.89170335664848572773611082696, −1.66613776482068750405307970845, 1.36176439626154840798600541290, 2.72917640650085355850502595296, 4.12970700547533590243962019265, 4.54756328129070407313444068607, 5.40751105900120137331485780759, 6.26569709280971825383608034719, 7.11072772033126657973261151717, 8.311414432946806849556596424751, 9.505268639137641021114800873694, 10.15566345599254900655540218805

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