Properties

Label 2-1176-21.20-c1-0-15
Degree $2$
Conductor $1176$
Sign $-0.0212 - 0.999i$
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  + (1.33 + 1.10i)3-s + 0.145·5-s + (0.554 + 2.94i)9-s + 2.46i·11-s + 2.04i·13-s + (0.193 + 0.160i)15-s + 1.75·17-s + 4.25i·19-s − 8.61i·23-s − 4.97·25-s + (−2.52 + 4.54i)27-s + 7.08i·29-s + 3.60i·31-s + (−2.73 + 3.29i)33-s + 5.86·37-s + ⋯
L(s)  = 1  + (0.769 + 0.638i)3-s + 0.0649·5-s + (0.184 + 0.982i)9-s + 0.744i·11-s + 0.566i·13-s + (0.0500 + 0.0414i)15-s + 0.426·17-s + 0.975i·19-s − 1.79i·23-s − 0.995·25-s + (−0.485 + 0.874i)27-s + 1.31i·29-s + 0.646i·31-s + (−0.475 + 0.573i)33-s + 0.964·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0212 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0212 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.0212 - 0.999i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -0.0212 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.025141601\)
\(L(\frac12)\) \(\approx\) \(2.025141601\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.33 - 1.10i)T \)
7 \( 1 \)
good5 \( 1 - 0.145T + 5T^{2} \)
11 \( 1 - 2.46iT - 11T^{2} \)
13 \( 1 - 2.04iT - 13T^{2} \)
17 \( 1 - 1.75T + 17T^{2} \)
19 \( 1 - 4.25iT - 19T^{2} \)
23 \( 1 + 8.61iT - 23T^{2} \)
29 \( 1 - 7.08iT - 29T^{2} \)
31 \( 1 - 3.60iT - 31T^{2} \)
37 \( 1 - 5.86T + 37T^{2} \)
41 \( 1 - 5.33T + 41T^{2} \)
43 \( 1 + 9.19T + 43T^{2} \)
47 \( 1 - 9.30T + 47T^{2} \)
53 \( 1 - 5.19iT - 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 5.38iT - 61T^{2} \)
67 \( 1 + 5.14T + 67T^{2} \)
71 \( 1 + 7.79iT - 71T^{2} \)
73 \( 1 - 13.0iT - 73T^{2} \)
79 \( 1 + 5.72T + 79T^{2} \)
83 \( 1 + 15.9T + 83T^{2} \)
89 \( 1 - 8.68T + 89T^{2} \)
97 \( 1 + 6.65iT - 97T^{2} \)
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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

−10.05095946264000005275079806088, −9.147754028414401552701986307254, −8.466274597680608506043780357076, −7.64622034046410347492587555773, −6.78569084882500786960687212181, −5.64231924876680616068958356636, −4.60791396688029708494334677876, −3.93144708126773136261035385503, −2.78799055998104537186108863408, −1.73669216396368492212237986956, 0.813390190067113229915887267655, 2.21540484601238893915785028089, 3.20412988564276855301305734539, 4.08179302203662506975274299159, 5.54854886404425524229364451906, 6.18179568829019870187775597447, 7.37231914863647820007900051611, 7.82510438970270180986444623803, 8.687910672457973754067547280059, 9.498405585704839526391694179431

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