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Properties

Label 2-1176-7.4-c1-0-13
Degree $2$
Conductor $1176$
Sign $0.0725 + 0.997i$
Analytic cond. $9.39040$
Root an. cond. $3.06437$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.292 − 0.507i)5-s + (−0.499 − 0.866i)9-s + (0.414 − 0.717i)11-s − 1.41·13-s + 0.585·15-s + (−1.12 + 1.94i)17-s + (−3.41 − 5.91i)19-s + (−2.41 − 4.18i)23-s + (2.32 − 4.03i)25-s + 0.999·27-s + 8.48·29-s + (−2.58 + 4.47i)31-s + (0.414 + 0.717i)33-s + (−0.828 − 1.43i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.130 − 0.226i)5-s + (−0.166 − 0.288i)9-s + (0.124 − 0.216i)11-s − 0.392·13-s + 0.151·15-s + (−0.271 + 0.471i)17-s + (−0.783 − 1.35i)19-s + (−0.503 − 0.871i)23-s + (0.465 − 0.806i)25-s + 0.192·27-s + 1.57·29-s + (−0.464 + 0.804i)31-s + (0.0721 + 0.124i)33-s + (−0.136 − 0.235i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.0725 + 0.997i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ 0.0725 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8904450776\)
\(L(\frac12)\) \(\approx\) \(0.8904450776\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (0.292 + 0.507i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.414 + 0.717i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 + (1.12 - 1.94i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.41 + 5.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.41 + 4.18i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.48T + 29T^{2} \)
31 \( 1 + (2.58 - 4.47i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.828 + 1.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.585T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (3.41 + 5.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.65 + 11.5i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.58 - 4.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.94 + 12.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.828T + 71T^{2} \)
73 \( 1 + (-5.53 + 9.58i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.17 - 2.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.3T + 83T^{2} \)
89 \( 1 + (-5.36 - 9.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.75T + 97T^{2} \)
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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

−9.628455065720313182047845672122, −8.649978673661570830294617149917, −8.266286779846722671100694672760, −6.79675697902159525923368402853, −6.40041235565133556863510268292, −5.03032590793203552306230410638, −4.57515403463125193615083984629, −3.42711906787468813352472343738, −2.22463118111153002588283762156, −0.40434148148258584559051177673, 1.43202623451119674778516661351, 2.63360262743405095762318986424, 3.84034382216470483185853148486, 4.89063098914215597149143541138, 5.87646974056209412381701561779, 6.65721104247207411393340652363, 7.48953194535882332033523808991, 8.168081876400701071213879833095, 9.156068882994246538721508748336, 10.05847487408354513076406572160

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