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Properties

Label 2-1176-56.27-c1-0-48
Degree $2$
Conductor $1176$
Sign $0.538 - 0.842i$
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  + (0.765 + 1.18i)2-s − i·3-s + (−0.828 + 1.82i)4-s + 4.17·5-s + (1.18 − 0.765i)6-s + (−2.79 + 0.407i)8-s − 9-s + (3.19 + 4.96i)10-s + 1.71·11-s + (1.82 + 0.828i)12-s + 1.54·13-s − 4.17i·15-s + (−2.62 − 3.01i)16-s − 2.33i·17-s + (−0.765 − 1.18i)18-s + 7.04i·19-s + ⋯
L(s)  = 1  + (0.541 + 0.840i)2-s − 0.577i·3-s + (−0.414 + 0.910i)4-s + 1.86·5-s + (0.485 − 0.312i)6-s + (−0.989 + 0.144i)8-s − 0.333·9-s + (1.01 + 1.57i)10-s + 0.515·11-s + (0.525 + 0.239i)12-s + 0.427·13-s − 1.07i·15-s + (−0.656 − 0.754i)16-s − 0.565i·17-s + (−0.180 − 0.280i)18-s + 1.61i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.538 - 0.842i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ 0.538 - 0.842i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.900404622\)
\(L(\frac12)\) \(\approx\) \(2.900404622\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.765 - 1.18i)T \)
3 \( 1 + iT \)
7 \( 1 \)
good5 \( 1 - 4.17T + 5T^{2} \)
11 \( 1 - 1.71T + 11T^{2} \)
13 \( 1 - 1.54T + 13T^{2} \)
17 \( 1 + 2.33iT - 17T^{2} \)
19 \( 1 - 7.04iT - 19T^{2} \)
23 \( 1 - 0.468iT - 23T^{2} \)
29 \( 1 - 3.33iT - 29T^{2} \)
31 \( 1 - 3.16T + 31T^{2} \)
37 \( 1 + 8.94iT - 37T^{2} \)
41 \( 1 + 5.31iT - 41T^{2} \)
43 \( 1 + 3.42T + 43T^{2} \)
47 \( 1 + 5.90T + 47T^{2} \)
53 \( 1 + 1.56iT - 53T^{2} \)
59 \( 1 - 6.08iT - 59T^{2} \)
61 \( 1 - 9.11T + 61T^{2} \)
67 \( 1 + 7.47T + 67T^{2} \)
71 \( 1 + 3.49iT - 71T^{2} \)
73 \( 1 - 14.5iT - 73T^{2} \)
79 \( 1 - 1.68iT - 79T^{2} \)
83 \( 1 - 2.72iT - 83T^{2} \)
89 \( 1 - 2.12iT - 89T^{2} \)
97 \( 1 + 1.95iT - 97T^{2} \)
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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

−9.674841710476821683555376399933, −8.996762415177237000742271372506, −8.219461498487626220283853718538, −7.11561216769435070476013277212, −6.45752243668670704589321559087, −5.75845791209259478472743648594, −5.23398520589589881109500230694, −3.83810418849464244882000453334, −2.62581539940560910569940304754, −1.49688844230620965316665564920, 1.27210105846788348868238872391, 2.35884077473888612108851450558, 3.24470774079113460504375529694, 4.56946257867462628983449501345, 5.16294777271592441070561294827, 6.22064447910411851369436238277, 6.53433298286073310773753960944, 8.501185822847722587127882749099, 9.197299768410065830602719736355, 9.802363240514228018380049370871

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