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Properties

Label 2-1176-21.20-c1-0-27
Degree $2$
Conductor $1176$
Sign $0.928 - 0.370i$
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  + (1.53 + 0.795i)3-s + 3.80·5-s + (1.73 + 2.44i)9-s + 0.357i·11-s − 4.04i·13-s + (5.84 + 3.02i)15-s + 0.103·17-s − 2.45i·19-s + 1.33i·23-s + 9.44·25-s + (0.723 + 5.14i)27-s − 4.97i·29-s + 7.88i·31-s + (−0.284 + 0.549i)33-s − 10.9·37-s + ⋯
L(s)  = 1  + (0.888 + 0.459i)3-s + 1.69·5-s + (0.578 + 0.815i)9-s + 0.107i·11-s − 1.12i·13-s + (1.50 + 0.780i)15-s + 0.0252·17-s − 0.563i·19-s + 0.277i·23-s + 1.88·25-s + (0.139 + 0.990i)27-s − 0.923i·29-s + 1.41i·31-s + (−0.0494 + 0.0957i)33-s − 1.79·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.928 - 0.370i$
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: \((2,\ 1176,\ (\ :1/2),\ 0.928 - 0.370i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.016209572\)
\(L(\frac12)\) \(\approx\) \(3.016209572\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.53 - 0.795i)T \)
7 \( 1 \)
good5 \( 1 - 3.80T + 5T^{2} \)
11 \( 1 - 0.357iT - 11T^{2} \)
13 \( 1 + 4.04iT - 13T^{2} \)
17 \( 1 - 0.103T + 17T^{2} \)
19 \( 1 + 2.45iT - 19T^{2} \)
23 \( 1 - 1.33iT - 23T^{2} \)
29 \( 1 + 4.97iT - 29T^{2} \)
31 \( 1 - 7.88iT - 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 - 6.15T + 41T^{2} \)
43 \( 1 - 0.502T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + 5.86iT - 53T^{2} \)
59 \( 1 + 7.54T + 59T^{2} \)
61 \( 1 - 9.47iT - 61T^{2} \)
67 \( 1 - 2.68T + 67T^{2} \)
71 \( 1 + 5.78iT - 71T^{2} \)
73 \( 1 - 0.235iT - 73T^{2} \)
79 \( 1 - 3.22T + 79T^{2} \)
83 \( 1 + 9.07T + 83T^{2} \)
89 \( 1 + 6.82T + 89T^{2} \)
97 \( 1 - 5.14iT - 97T^{2} \)
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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

−9.883424254884937497667867575001, −9.075588431981151126369275436680, −8.437551826138433732865788204724, −7.41205906740833663153355727774, −6.44043473967465434326988234329, −5.44586559953150012675733648074, −4.82843056023784145571633929821, −3.38776894810991331498677475241, −2.55813065430239150441286509236, −1.55428600586531301866859443455, 1.52481199038553218980066214901, 2.16143583030075006172444562614, 3.27085811635819892588758769977, 4.51000594705930415865037406931, 5.69005822190025802067887098028, 6.44788374695092912811994613101, 7.12169196792844273783837931183, 8.242821143914759198767690168731, 9.058531033671187539648835223349, 9.555996113749053888489553769344

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