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Properties

Label 2-912-19.12-c2-0-6
Degree $2$
Conductor $912$
Sign $-0.696 - 0.717i$
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 + (0.531 + 0.921i)5-s + 2.38·7-s + (1.5 − 2.59i)9-s − 6.99·11-s + (7.15 + 4.12i)13-s + (−1.59 − 0.921i)15-s + (2.12 + 3.68i)17-s + (−17.8 + 6.43i)19-s + (−3.57 + 2.06i)21-s + (−11.0 + 19.1i)23-s + (11.9 − 20.6i)25-s + 5.19i·27-s + (30.8 + 17.8i)29-s − 21.7i·31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (0.106 + 0.184i)5-s + 0.340·7-s + (0.166 − 0.288i)9-s − 0.636·11-s + (0.550 + 0.317i)13-s + (−0.106 − 0.0614i)15-s + (0.125 + 0.216i)17-s + (−0.940 + 0.338i)19-s + (−0.170 + 0.0983i)21-s + (−0.481 + 0.834i)23-s + (0.477 − 0.826i)25-s + 0.192i·27-s + (1.06 + 0.614i)29-s − 0.701i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.696 - 0.717i$
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: \((2,\ 912,\ (\ :1),\ -0.696 - 0.717i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9137812158\)
\(L(\frac12)\) \(\approx\) \(0.9137812158\)
\(L(2)\) 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 \( 1 + (17.8 - 6.43i)T \)
good5 \( 1 + (-0.531 - 0.921i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 2.38T + 49T^{2} \)
11 \( 1 + 6.99T + 121T^{2} \)
13 \( 1 + (-7.15 - 4.12i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-2.12 - 3.68i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (11.0 - 19.1i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-30.8 - 17.8i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 21.7iT - 961T^{2} \)
37 \( 1 - 20.4iT - 1.36e3T^{2} \)
41 \( 1 + (-2.96 + 1.71i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-11.1 - 19.2i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (23.6 - 40.9i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (21.8 + 12.5i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (75.5 - 43.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (27.4 - 47.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (12.2 + 7.09i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (23.7 - 13.7i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-31.9 - 55.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-2.10 + 1.21i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 40.4T + 6.88e3T^{2} \)
89 \( 1 + (30.2 + 17.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (72.9 - 42.1i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.37290693893158259767714969193, −9.499977054504412718328741438829, −8.465744679514025990237670976507, −7.78466179340765783670329703982, −6.57359934818835633447134792011, −5.96105094276647818583068192538, −4.89289159771412881415647533263, −4.07778084553110857276781166172, −2.80490439181702578854030252006, −1.41136856751989057664735550766, 0.32223379844606971762060861927, 1.72194901970184442466708492986, 2.98481795478980580123929512723, 4.40652690380761970621532527422, 5.18559952556382026064856370684, 6.13112693673343136884174803410, 6.92493573860283395674368089119, 8.001140555423494072970778106763, 8.565482209545921077128745880164, 9.657286351285703199968249741345

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