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Properties

Label 2-1176-21.20-c1-0-29
Degree $2$
Conductor $1176$
Sign $0.956 + 0.290i$
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  + (1.71 + 0.238i)3-s + 1.32·5-s + (2.88 + 0.817i)9-s − 2.81i·11-s − 2.52i·13-s + (2.27 + 0.315i)15-s + 3.26·17-s − 5.73i·19-s − 4.25i·23-s − 3.24·25-s + (4.75 + 2.09i)27-s + 4.75i·29-s + 9.10i·31-s + (0.669 − 4.82i)33-s + 11.2·37-s + ⋯
L(s)  = 1  + (0.990 + 0.137i)3-s + 0.592·5-s + (0.962 + 0.272i)9-s − 0.847i·11-s − 0.701i·13-s + (0.586 + 0.0814i)15-s + 0.791·17-s − 1.31i·19-s − 0.887i·23-s − 0.649·25-s + (0.915 + 0.402i)27-s + 0.883i·29-s + 1.63i·31-s + (0.116 − 0.839i)33-s + 1.85·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.956 + 0.290i$
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: \((2,\ 1176,\ (\ :1/2),\ 0.956 + 0.290i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.667373979\)
\(L(\frac12)\) \(\approx\) \(2.667373979\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.238i)T \)
7 \( 1 \)
good5 \( 1 - 1.32T + 5T^{2} \)
11 \( 1 + 2.81iT - 11T^{2} \)
13 \( 1 + 2.52iT - 13T^{2} \)
17 \( 1 - 3.26T + 17T^{2} \)
19 \( 1 + 5.73iT - 19T^{2} \)
23 \( 1 + 4.25iT - 23T^{2} \)
29 \( 1 - 4.75iT - 29T^{2} \)
31 \( 1 - 9.10iT - 31T^{2} \)
37 \( 1 - 11.2T + 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 - 9.48T + 43T^{2} \)
47 \( 1 + 1.48T + 47T^{2} \)
53 \( 1 - 12.0iT - 53T^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 + 2.34iT - 61T^{2} \)
67 \( 1 + 0.130T + 67T^{2} \)
71 \( 1 - 0.875iT - 71T^{2} \)
73 \( 1 - 6.44iT - 73T^{2} \)
79 \( 1 - 5.93T + 79T^{2} \)
83 \( 1 + 0.398T + 83T^{2} \)
89 \( 1 + 5.75T + 89T^{2} \)
97 \( 1 - 5.36iT - 97T^{2} \)
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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

−9.597953276613385794045010735716, −8.928700573927394611177845683273, −8.229675267414690608146540328830, −7.40648356656806793512795011043, −6.43834116842103955579179968089, −5.45225188978231209105729255437, −4.50142194522914773019158148314, −3.24901180716527282892262441525, −2.64467060857497976998519801968, −1.19089643894046429373155007814, 1.57480174789616319405796761847, 2.33755707268523996245838970395, 3.64045057982640857332828740894, 4.40161625439740929727259888386, 5.70220180059322607295041069698, 6.49594027020944637043494226080, 7.73478132008010929908308278399, 7.87166883576451156700819944037, 9.247304053457326928352893233498, 9.698140128210027557503829259200

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