Properties

Label 2-1176-21.20-c1-0-34
Degree $2$
Conductor $1176$
Sign $-0.188 + 0.982i$
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  + (1.07 + 1.36i)3-s − 2.57·5-s + (−0.703 + 2.91i)9-s − 1.65i·11-s − 5.71i·13-s + (−2.76 − 3.50i)15-s − 7.58·17-s − 2.99i·19-s + 0.287i·23-s + 1.65·25-s + (−4.72 + 2.16i)27-s + 2.05i·29-s − 6.01i·31-s + (2.25 − 1.77i)33-s + 1.75·37-s + ⋯
L(s)  = 1  + (0.618 + 0.785i)3-s − 1.15·5-s + (−0.234 + 0.972i)9-s − 0.498i·11-s − 1.58i·13-s + (−0.713 − 0.906i)15-s − 1.83·17-s − 0.686i·19-s + 0.0600i·23-s + 0.330·25-s + (−0.908 + 0.417i)27-s + 0.382i·29-s − 1.08i·31-s + (0.391 − 0.308i)33-s + 0.288·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.188 + 0.982i$
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: \((2,\ 1176,\ (\ :1/2),\ -0.188 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6247549316\)
\(L(\frac12)\) \(\approx\) \(0.6247549316\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.07 - 1.36i)T \)
7 \( 1 \)
good5 \( 1 + 2.57T + 5T^{2} \)
11 \( 1 + 1.65iT - 11T^{2} \)
13 \( 1 + 5.71iT - 13T^{2} \)
17 \( 1 + 7.58T + 17T^{2} \)
19 \( 1 + 2.99iT - 19T^{2} \)
23 \( 1 - 0.287iT - 23T^{2} \)
29 \( 1 - 2.05iT - 29T^{2} \)
31 \( 1 + 6.01iT - 31T^{2} \)
37 \( 1 - 1.75T + 37T^{2} \)
41 \( 1 - 4.28T + 41T^{2} \)
43 \( 1 - 2.46T + 43T^{2} \)
47 \( 1 - 0.373T + 47T^{2} \)
53 \( 1 + 7.77iT - 53T^{2} \)
59 \( 1 + 9.79T + 59T^{2} \)
61 \( 1 + 1.02iT - 61T^{2} \)
67 \( 1 - 2.36T + 67T^{2} \)
71 \( 1 + 15.6iT - 71T^{2} \)
73 \( 1 + 3.81iT - 73T^{2} \)
79 \( 1 + 9.12T + 79T^{2} \)
83 \( 1 + 6.65T + 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 - 4.43iT - 97T^{2} \)
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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

−9.397747012681930927191249205203, −8.692045822300714637392825865752, −8.005733207378918220837582853470, −7.39314817936096859000370122362, −6.12368361621359903462572300620, −4.99361717895032256698741831679, −4.22337898626155630214626649521, −3.37370233133169181429097556070, −2.48519886739250895073206046087, −0.24084347121882866002917069710, 1.61884371466647499732910251965, 2.67612078872750795201390725916, 4.02648415121128944824047003420, 4.42344652270869749479878327006, 6.10961587235294539382506171479, 6.96960777716751380175310718133, 7.40248677821904675690075110533, 8.436584293990812896055045213769, 8.908879599336817414362622730738, 9.785249989056849364817876488702

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