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Properties

Label 2-912-19.12-c2-0-34
Degree $2$
Conductor $912$
Sign $-0.859 + 0.511i$
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 + (3.80 + 6.59i)5-s − 10.8·7-s + (1.5 − 2.59i)9-s − 9.11·11-s + (4.51 + 2.60i)13-s + (11.4 + 6.59i)15-s + (−13.2 − 22.8i)17-s + (−2.11 − 18.8i)19-s + (−16.3 + 9.42i)21-s + (−20.1 + 34.9i)23-s + (−16.5 + 28.6i)25-s − 5.19i·27-s + (12.7 + 7.34i)29-s − 36.3i·31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (0.761 + 1.31i)5-s − 1.55·7-s + (0.166 − 0.288i)9-s − 0.828·11-s + (0.347 + 0.200i)13-s + (0.761 + 0.439i)15-s + (−0.777 − 1.34i)17-s + (−0.111 − 0.993i)19-s + (−0.777 + 0.448i)21-s + (−0.877 + 1.51i)23-s + (−0.660 + 1.14i)25-s − 0.192i·27-s + (0.438 + 0.253i)29-s − 1.17i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.859 + 0.511i$
Analytic conductor: \(24.8502\)
Root analytic conductor: \(4.98499\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1),\ -0.859 + 0.511i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2343316410\)
\(L(\frac12)\) \(\approx\) \(0.2343316410\)
\(L(2)\) 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 \( 1 + (2.11 + 18.8i)T \)
good5 \( 1 + (-3.80 - 6.59i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 10.8T + 49T^{2} \)
11 \( 1 + 9.11T + 121T^{2} \)
13 \( 1 + (-4.51 - 2.60i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (13.2 + 22.8i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (20.1 - 34.9i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-12.7 - 7.34i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 36.3iT - 961T^{2} \)
37 \( 1 + 43.6iT - 1.36e3T^{2} \)
41 \( 1 + (48.5 - 28.0i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (23.5 + 40.7i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-15.2 + 26.3i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (21.1 + 12.2i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (2.34 - 1.35i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-3.41 + 5.91i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (3.82 + 2.20i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (96.2 - 55.5i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-24.4 - 42.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-50.9 + 29.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 151.T + 6.88e3T^{2} \)
89 \( 1 + (129. + 74.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (88.7 - 51.2i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.628470941149426053892693141276, −8.971982231294567243693638386950, −7.59135353913939287851215900411, −6.89361261260200927231637919043, −6.36813699123491374851105627190, −5.37718661924191364451505233243, −3.72251307070344996130019766401, −2.86012517340385276201410534202, −2.24949457930784294212765725685, −0.06394188678846317865396976122, 1.61837791572209925467232345162, 2.84694386269449714446382731586, 3.96052008970290801380087862450, 4.92314381496972939613142620201, 6.01624508957678145743404046688, 6.53364281865921860454107057435, 8.195610360574279956678012104887, 8.521520954996531372095032090113, 9.421144934078081035871513842409, 10.20585950080704478118586578050

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