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Properties

Label 2-912-57.2-c1-0-4
Degree $2$
Conductor $912$
Sign $-0.149 - 0.988i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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  + (−1.67 − 0.431i)3-s + (1.14 + 3.13i)5-s + (1.07 + 1.85i)7-s + (2.62 + 1.44i)9-s + (5.41 + 3.12i)11-s + (−2.56 − 3.05i)13-s + (−0.560 − 5.75i)15-s + (−0.403 − 0.0711i)17-s + (−4.34 + 0.329i)19-s + (−0.998 − 3.58i)21-s + (0.280 − 0.770i)23-s + (−4.70 + 3.94i)25-s + (−3.78 − 3.56i)27-s + (0.805 + 4.56i)29-s + (2.02 − 1.16i)31-s + ⋯
L(s)  = 1  + (−0.968 − 0.249i)3-s + (0.510 + 1.40i)5-s + (0.405 + 0.702i)7-s + (0.875 + 0.482i)9-s + (1.63 + 0.943i)11-s + (−0.710 − 0.846i)13-s + (−0.144 − 1.48i)15-s + (−0.0978 − 0.0172i)17-s + (−0.997 + 0.0756i)19-s + (−0.217 − 0.781i)21-s + (0.0584 − 0.160i)23-s + (−0.940 + 0.788i)25-s + (−0.727 − 0.685i)27-s + (0.149 + 0.847i)29-s + (0.363 − 0.210i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.149 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.149 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.149 - 0.988i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.149 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.820408 + 0.953478i\)
\(L(\frac12)\) \(\approx\) \(0.820408 + 0.953478i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 + 0.431i)T \)
19 \( 1 + (4.34 - 0.329i)T \)
good5 \( 1 + (-1.14 - 3.13i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (-1.07 - 1.85i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.41 - 3.12i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.56 + 3.05i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.403 + 0.0711i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (-0.280 + 0.770i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.805 - 4.56i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-2.02 + 1.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.01iT - 37T^{2} \)
41 \( 1 + (-0.926 - 0.777i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (5.87 - 2.13i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-7.59 + 1.33i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (0.220 + 0.0802i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.930 - 5.27i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-7.30 - 2.65i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (3.48 - 0.614i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (4.19 - 1.52i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (4.33 + 3.63i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-8.05 + 9.59i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (8.01 - 4.62i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.61 - 4.71i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (16.0 + 2.83i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.31598205996641814353079859252, −9.835223488133703637827278134210, −8.696290194093180020002691239554, −7.43565619065616007280717800941, −6.73800955090495267292271989124, −6.21660611959730208228583615319, −5.21743277868598702946596079953, −4.18176102312032742878730994683, −2.68551147658121075157963482804, −1.66176454960513650081467805965, 0.72127453340206038960923757390, 1.71441062276507269681973199325, 4.06615788132399322066156777439, 4.41433066556927577706766156784, 5.46577642687873247125901513666, 6.29694133402903016210106119864, 7.08545147270981352162987964599, 8.419659920472364138936205686043, 9.151386141159966825177898141932, 9.747639339839443253009098500035

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