Properties

Label 2-912-19.12-c2-0-20
Degree $2$
Conductor $912$
Sign $0.412 + 0.910i$
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 + 1.73i)5-s + 7-s + (1.5 − 2.59i)9-s − 16·11-s + (13.5 + 7.79i)13-s + (−3 − 1.73i)15-s + (−11 − 19.0i)17-s − 19·19-s + (−1.5 + 0.866i)21-s + (20 − 34.6i)23-s + (10.5 − 18.1i)25-s + 5.19i·27-s + (15 + 8.66i)29-s + 50.2i·31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (0.200 + 0.346i)5-s + 0.142·7-s + (0.166 − 0.288i)9-s − 1.45·11-s + (1.03 + 0.599i)13-s + (−0.200 − 0.115i)15-s + (−0.647 − 1.12i)17-s − 19-s + (−0.0714 + 0.0412i)21-s + (0.869 − 1.50i)23-s + (0.419 − 0.727i)25-s + 0.192i·27-s + (0.517 + 0.298i)29-s + 1.62i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.412 + 0.910i$
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: \((2,\ 912,\ (\ :1),\ 0.412 + 0.910i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.053594361\)
\(L(\frac12)\) \(\approx\) \(1.053594361\)
\(L(2)\) 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 \( 1 + 19T \)
good5 \( 1 + (-1 - 1.73i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - T + 49T^{2} \)
11 \( 1 + 16T + 121T^{2} \)
13 \( 1 + (-13.5 - 7.79i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (11 + 19.0i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-20 + 34.6i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-15 - 8.66i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 50.2iT - 961T^{2} \)
37 \( 1 - 15.5iT - 1.36e3T^{2} \)
41 \( 1 + (12 - 6.92i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (24.5 + 42.4i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-23 + 39.8i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (42 + 24.2i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-57 + 32.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-48.5 + 84.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-22.5 - 12.9i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-42 + 24.2i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (17.5 + 30.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-76.5 + 44.1i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 146T + 6.88e3T^{2} \)
89 \( 1 + (33 + 19.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-54 + 31.1i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.974818738230425908094484805558, −8.767915940042164497238343812642, −8.297527240441737650583372561174, −6.76004682635692444977232427449, −6.58195562829461375867208906795, −5.13329655323861776094303914536, −4.67535189614147769874615993908, −3.24806116852241526100197985956, −2.17894078228272399717872110227, −0.40289497913648998223260979410, 1.13856745598346454833444355645, 2.40105091743757672843814327720, 3.76998426192481984713727362576, 4.95156242445396215273711041273, 5.70980782495832438522797120028, 6.44528218927577536328470491133, 7.67798130070173116229662946795, 8.224230624011749101885159893770, 9.162395086499320923245610504022, 10.24663271245405403042240815131

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