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Properties

Label 2-1176-1176.533-c0-0-1
Degree $2$
Conductor $1176$
Sign $-0.0960 - 0.995i$
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  + (0.623 − 0.781i)2-s + (−0.900 − 0.433i)3-s + (−0.222 − 0.974i)4-s + (−1.12 − 0.541i)5-s + (−0.900 + 0.433i)6-s + (−0.900 + 0.433i)7-s + (−0.900 − 0.433i)8-s + (0.623 + 0.781i)9-s + (−1.12 + 0.541i)10-s + (−0.277 + 0.347i)11-s + (−0.222 + 0.974i)12-s + (−0.222 + 0.974i)14-s + (0.777 + 0.974i)15-s + (−0.900 + 0.433i)16-s + 18-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.900 − 0.433i)3-s + (−0.222 − 0.974i)4-s + (−1.12 − 0.541i)5-s + (−0.900 + 0.433i)6-s + (−0.900 + 0.433i)7-s + (−0.900 − 0.433i)8-s + (0.623 + 0.781i)9-s + (−1.12 + 0.541i)10-s + (−0.277 + 0.347i)11-s + (−0.222 + 0.974i)12-s + (−0.222 + 0.974i)14-s + (0.777 + 0.974i)15-s + (−0.900 + 0.433i)16-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0960 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0960 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.0960 - 0.995i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ -0.0960 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08736413515\)
\(L(\frac12)\) \(\approx\) \(0.08736413515\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.623 + 0.781i)T \)
3 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (0.900 - 0.433i)T \)
good5 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
11 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
13 \( 1 + (0.222 + 0.974i)T^{2} \)
17 \( 1 + (0.900 + 0.433i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.900 - 0.433i)T^{2} \)
29 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
31 \( 1 + 1.80T + T^{2} \)
37 \( 1 + (0.900 + 0.433i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.445 + 1.94i)T + (-0.900 + 0.433i)T^{2} \)
59 \( 1 + (1.80 - 0.867i)T + (0.623 - 0.781i)T^{2} \)
61 \( 1 + (0.900 + 0.433i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
79 \( 1 + 1.80T + T^{2} \)
83 \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.222 - 0.974i)T^{2} \)
97 \( 1 + 0.445T + T^{2} \)
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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

−9.544892738448915738080920358262, −8.709346471705955633629035500129, −7.53140914998318925368785271278, −6.74378767007787960120172109797, −5.74799558860765209417712241318, −5.04261713519169634026814022431, −4.13442804599470177122820529409, −3.17267122554581779714080885191, −1.73475326122063321593680005530, −0.06851505592944514004000200950, 3.08843554310622611091191879486, 3.81683934920932558574789314126, 4.49185157645943920122146376898, 5.68843691522698953680523399443, 6.29485524478870770108173531260, 7.26380822269794891398164568092, 7.62549813323879311341836212244, 8.891092282298045647007144141602, 9.759227319809836276713423549841, 10.81330404060048118484318462213

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