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Properties

Label 2-1176-8.5-c1-0-62
Degree $2$
Conductor $1176$
Sign $0.570 + 0.821i$
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  + (1.38 − 0.284i)2-s − i·3-s + (1.83 − 0.788i)4-s + 2.29i·5-s + (−0.284 − 1.38i)6-s + (2.32 − 1.61i)8-s − 9-s + (0.652 + 3.17i)10-s − 3.88i·11-s + (−0.788 − 1.83i)12-s − 3.33i·13-s + 2.29·15-s + (2.75 − 2.89i)16-s + 0.287·17-s + (−1.38 + 0.284i)18-s + 2.78i·19-s + ⋯
L(s)  = 1  + (0.979 − 0.201i)2-s − 0.577i·3-s + (0.919 − 0.394i)4-s + 1.02i·5-s + (−0.116 − 0.565i)6-s + (0.821 − 0.570i)8-s − 0.333·9-s + (0.206 + 1.00i)10-s − 1.17i·11-s + (−0.227 − 0.530i)12-s − 0.924i·13-s + 0.592·15-s + (0.689 − 0.724i)16-s + 0.0696·17-s + (−0.326 + 0.0670i)18-s + 0.638i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.570 + 0.821i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ 0.570 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.159113199\)
\(L(\frac12)\) \(\approx\) \(3.159113199\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.284i)T \)
3 \( 1 + iT \)
7 \( 1 \)
good5 \( 1 - 2.29iT - 5T^{2} \)
11 \( 1 + 3.88iT - 11T^{2} \)
13 \( 1 + 3.33iT - 13T^{2} \)
17 \( 1 - 0.287T + 17T^{2} \)
19 \( 1 - 2.78iT - 19T^{2} \)
23 \( 1 - 6.53T + 23T^{2} \)
29 \( 1 + 5.53iT - 29T^{2} \)
31 \( 1 - 7.44T + 31T^{2} \)
37 \( 1 + 5.95iT - 37T^{2} \)
41 \( 1 + 3.51T + 41T^{2} \)
43 \( 1 - 11.2iT - 43T^{2} \)
47 \( 1 - 0.0870T + 47T^{2} \)
53 \( 1 - 7.05iT - 53T^{2} \)
59 \( 1 - 4.35iT - 59T^{2} \)
61 \( 1 - 7.16iT - 61T^{2} \)
67 \( 1 - 13.0iT - 67T^{2} \)
71 \( 1 + 6.18T + 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 + 8.99T + 79T^{2} \)
83 \( 1 + 17.6iT - 83T^{2} \)
89 \( 1 + 17.1T + 89T^{2} \)
97 \( 1 - 6.46T + 97T^{2} \)
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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

−10.05731180533401289451572450174, −8.653725498413360988664545888028, −7.71536665416778399533912934490, −7.02727572644110572327724388268, −6.08373128480821660270033953207, −5.69059851095762075778364390229, −4.35617049502888137394547896143, −3.03866280580582377963606065555, −2.82640320801382722441521484157, −1.12084594579307544412662433524, 1.58221013512802512833152053653, 2.88760348580247676857931270739, 4.08331482637001121531241446654, 4.89743067841845845555322476567, 5.12911828299670014468759282368, 6.58652243562235685294951630444, 7.12684501103480246191171871086, 8.347947238876001265953811879296, 9.021936369990172342351499164659, 9.896654557714500448986387941284

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