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Properties

Label 2-1176-1176.533-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(1.140776687\)
\(L(\frac12)\) \(\approx\) \(1.140776687\)
\(L(1)\) 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

−10.05173695047610444108666415141, −9.172790473355728938634833578750, −8.890711288661201146068818641069, −7.75519146324934440685066080576, −6.91706654469895728911377271760, −6.07756479210874138186756906567, −5.45700068178031884221034257067, −4.09682720471812270540734987401, −2.84772615290670103722587983856, −1.89597981371614861146131952276, 1.29501158029062519268834867645, 2.21466047674134439460304380160, 3.27834126531064964019749272184, 4.14153564345832729697512610066, 5.52949983953321965008463475919, 6.84131876840728597962773116028, 7.26823714896287336488610245987, 8.563794539733681462026703934742, 8.975283857226803259267932462257, 9.860756189850507013808702184798

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