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Properties

Label 2-912-19.8-c2-0-8
Degree $2$
Conductor $912$
Sign $-0.971 - 0.238i$
Analytic cond. $24.8502$
Root an. cond. $4.98499$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 0.866i)3-s + (−1.41 + 2.44i)5-s − 2.35·7-s + (1.5 + 2.59i)9-s + 6.75·11-s + (−5.81 + 3.35i)13-s + (−4.23 + 2.44i)15-s + (−14.4 + 25.0i)17-s + (−3.49 − 18.6i)19-s + (−3.52 − 2.03i)21-s + (−10.0 − 17.4i)23-s + (8.52 + 14.7i)25-s + 5.19i·27-s + (−21.4 + 12.3i)29-s + 2.95i·31-s + ⋯
L(s)  = 1  + (0.5 + 0.288i)3-s + (−0.282 + 0.488i)5-s − 0.335·7-s + (0.166 + 0.288i)9-s + 0.614·11-s + (−0.447 + 0.258i)13-s + (−0.282 + 0.162i)15-s + (−0.852 + 1.47i)17-s + (−0.183 − 0.982i)19-s + (−0.167 − 0.0969i)21-s + (−0.436 − 0.756i)23-s + (0.340 + 0.590i)25-s + 0.192i·27-s + (−0.738 + 0.426i)29-s + 0.0954i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.971 - 0.238i$
Analytic conductor: \(24.8502\)
Root analytic conductor: \(4.98499\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1),\ -0.971 - 0.238i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8454710218\)
\(L(\frac12)\) \(\approx\) \(0.8454710218\)
\(L(2)\) 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 \( 1 + (-1.5 - 0.866i)T \)
19 \( 1 + (3.49 + 18.6i)T \)
good5 \( 1 + (1.41 - 2.44i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + 2.35T + 49T^{2} \)
11 \( 1 - 6.75T + 121T^{2} \)
13 \( 1 + (5.81 - 3.35i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (14.4 - 25.0i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (10.0 + 17.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (21.4 - 12.3i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 2.95iT - 961T^{2} \)
37 \( 1 + 18.6iT - 1.36e3T^{2} \)
41 \( 1 + (2.27 + 1.31i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (22.7 - 39.4i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (6.93 + 12.0i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (53.8 - 31.1i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-38.4 - 22.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (23.4 + 40.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (100. - 58.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (16.3 + 9.43i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-16.0 + 27.8i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (95.8 + 55.3i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 43.1T + 6.88e3T^{2} \)
89 \( 1 + (-26.7 + 15.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-33.9 - 19.5i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.33962966863036001735871868277, −9.290574594983979247638593314710, −8.785542106092850341808106592997, −7.77390612939915960227056879260, −6.84767063410944429616489514898, −6.19556022697559584148787852780, −4.76810193124437887736241164596, −3.93674736618948563618262449792, −2.98857801430246056974404006898, −1.80893131289660265760507515113, 0.24123944636559081948350895661, 1.74132362186798608718608667637, 2.98434340768286780484732593545, 4.03005588386186753336879434595, 4.98572244283426308855055617550, 6.15586168542000530047258843159, 7.05572151819048076787457348311, 7.85048988071209458817191933399, 8.685244441681198097622086293187, 9.460961910502742673358304554326

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