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Properties

Label 2-1176-1176.365-c0-0-1
Degree $2$
Conductor $1176$
Sign $0.159 + 0.987i$
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  + (0.900 − 0.433i)2-s + (0.222 − 0.974i)3-s + (0.623 − 0.781i)4-s + (−0.400 + 1.75i)5-s + (−0.222 − 0.974i)6-s + (−0.222 − 0.974i)7-s + (0.222 − 0.974i)8-s + (−0.900 − 0.433i)9-s + (0.400 + 1.75i)10-s + (1.12 − 0.541i)11-s + (−0.623 − 0.781i)12-s + (−0.623 − 0.781i)14-s + (1.62 + 0.781i)15-s + (−0.222 − 0.974i)16-s − 18-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)2-s + (0.222 − 0.974i)3-s + (0.623 − 0.781i)4-s + (−0.400 + 1.75i)5-s + (−0.222 − 0.974i)6-s + (−0.222 − 0.974i)7-s + (0.222 − 0.974i)8-s + (−0.900 − 0.433i)9-s + (0.400 + 1.75i)10-s + (1.12 − 0.541i)11-s + (−0.623 − 0.781i)12-s + (−0.623 − 0.781i)14-s + (1.62 + 0.781i)15-s + (−0.222 − 0.974i)16-s − 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.159 + 0.987i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ 0.159 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.764072945\)
\(L(\frac12)\) \(\approx\) \(1.764072945\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.900 + 0.433i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
7 \( 1 + (0.222 + 0.974i)T \)
good5 \( 1 + (0.400 - 1.75i)T + (-0.900 - 0.433i)T^{2} \)
11 \( 1 + (-1.12 + 0.541i)T + (0.623 - 0.781i)T^{2} \)
13 \( 1 + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + (0.222 - 0.974i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.222 + 0.974i)T^{2} \)
29 \( 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2} \)
31 \( 1 + 0.445T + T^{2} \)
37 \( 1 + (0.222 - 0.974i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (1.24 - 1.56i)T + (-0.222 - 0.974i)T^{2} \)
59 \( 1 + (-0.445 - 1.94i)T + (-0.900 + 0.433i)T^{2} \)
61 \( 1 + (0.222 - 0.974i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
79 \( 1 + 0.445T + T^{2} \)
83 \( 1 + (1.62 + 0.781i)T + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.623 - 0.781i)T^{2} \)
97 \( 1 - 1.24T + T^{2} \)
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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

−10.19912880842338350344514488691, −9.021770906668554052093881989156, −7.69864935282585800710860223790, −6.99821190421249629355295340137, −6.60942722867441591147090037492, −5.87452214780547641670780890315, −4.21879987962988632270725381513, −3.35987201345848824916933306523, −2.82249190157994458866392435801, −1.36938902529244180024157100912, 2.04252862584074954238038110875, 3.46064354518492941679448942588, 4.31683492103487372814274744394, 4.88436427007875037298899439848, 5.63199425844512818719117637335, 6.53775436406387354941048069246, 8.008007412933268570497536047526, 8.467005671810575955790678800733, 9.235658633777710356076363231917, 9.818282250103247269262100314631

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