Properties

Label 2-912-19.8-c2-0-6
Degree $2$
Conductor $912$
Sign $-0.940 - 0.340i$
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 + (−2.40 + 4.16i)5-s − 2.71·7-s + (1.5 + 2.59i)9-s + 2.72·11-s + (9.25 − 5.34i)13-s + (7.20 − 4.16i)15-s + (5.69 − 9.86i)17-s + (1.47 + 18.9i)19-s + (4.06 + 2.34i)21-s + (1.53 + 2.65i)23-s + (0.962 + 1.66i)25-s − 5.19i·27-s + (−15.7 + 9.11i)29-s + 20.6i·31-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (−0.480 + 0.832i)5-s − 0.387·7-s + (0.166 + 0.288i)9-s + 0.248·11-s + (0.711 − 0.410i)13-s + (0.480 − 0.277i)15-s + (0.335 − 0.580i)17-s + (0.0776 + 0.996i)19-s + (0.193 + 0.111i)21-s + (0.0665 + 0.115i)23-s + (0.0384 + 0.0666i)25-s − 0.192i·27-s + (−0.544 + 0.314i)29-s + 0.665i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.940 - 0.340i$
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: \((2,\ 912,\ (\ :1),\ -0.940 - 0.340i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3942052992\)
\(L(\frac12)\) \(\approx\) \(0.3942052992\)
\(L(2)\) 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 \( 1 + (-1.47 - 18.9i)T \)
good5 \( 1 + (2.40 - 4.16i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + 2.71T + 49T^{2} \)
11 \( 1 - 2.72T + 121T^{2} \)
13 \( 1 + (-9.25 + 5.34i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-5.69 + 9.86i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (-1.53 - 2.65i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (15.7 - 9.11i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 20.6iT - 961T^{2} \)
37 \( 1 + 45.7iT - 1.36e3T^{2} \)
41 \( 1 + (24.4 + 14.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (32.2 - 55.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (34.1 + 59.1i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (51.8 - 29.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (14.6 + 8.47i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-13.6 - 23.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-25.0 + 14.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (43.1 + 24.8i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (55.5 - 96.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (47.5 + 27.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 2.79T + 6.88e3T^{2} \)
89 \( 1 + (142. - 82.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-79.8 - 46.1i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.39180508499984347074425208934, −9.587574099571467386734246382257, −8.466926903898805558485809238966, −7.57211743763675896945606752229, −6.86693844171918287335340130560, −6.06721473188759777500412162394, −5.16236069754845107701271868823, −3.77295354778744191538582473122, −3.06245901510593223851176809612, −1.45699781972027698465739487078, 0.14612279639245242176576496109, 1.46134574246074604969637822501, 3.24049835057699632272606592484, 4.24716681643804907380507317057, 4.97143881472601194553433111452, 6.07840430008908219493579095896, 6.78694716049771741796991578007, 7.989868935569411103990722608159, 8.731153807176590068989232914064, 9.503767327690935198750146977756

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