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Properties

Label 2-882-21.11-c2-0-22
Degree $2$
Conductor $882$
Sign $0.216 + 0.976i$
Analytic cond. $24.0327$
Root an. cond. $4.90232$
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  + (1.22 − 0.707i)2-s + (0.999 − 1.73i)4-s + (7.44 − 4.30i)5-s − 2.82i·8-s + (6.08 − 10.5i)10-s + (2.44 + 1.41i)11-s + 12.1·13-s + (−2.00 − 3.46i)16-s + (−22.3 − 12.9i)17-s + (12.1 + 21.0i)19-s − 17.2i·20-s + 4·22-s + (36.7 − 21.2i)23-s + (24.5 − 42.4i)25-s + (14.8 − 8.60i)26-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (1.48 − 0.860i)5-s − 0.353i·8-s + (0.608 − 1.05i)10-s + (0.222 + 0.128i)11-s + 0.935·13-s + (−0.125 − 0.216i)16-s + (−1.31 − 0.759i)17-s + (0.640 + 1.10i)19-s − 0.860i·20-s + 0.181·22-s + (1.59 − 0.922i)23-s + (0.979 − 1.69i)25-s + (0.573 − 0.330i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.216 + 0.976i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ 0.216 + 0.976i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.728949075\)
\(L(\frac12)\) \(\approx\) \(3.728949075\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-7.44 + 4.30i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.44 - 1.41i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 12.1T + 169T^{2} \)
17 \( 1 + (22.3 + 12.9i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-12.1 - 21.0i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-36.7 + 21.2i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 15.5iT - 841T^{2} \)
31 \( 1 + (12.1 - 21.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 25.8iT - 1.68e3T^{2} \)
43 \( 1 + 68T + 1.84e3T^{2} \)
47 \( 1 + (59.5 - 34.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-35.5 - 20.5i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-59.5 - 34.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (48.6 + 84.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-52 + 90.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 70.7iT - 5.04e3T^{2} \)
73 \( 1 + (30.4 - 52.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-10 - 17.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 + (-81.9 + 47.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 158.T + 9.40e3T^{2} \)
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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

−9.714258199744815125856134992347, −9.126668441541410361134449128850, −8.328290009733941752399254260997, −6.77095182459540786516084066456, −6.18811121302689644462560633022, −5.19305477952705675760175470220, −4.61622469748562177024958970045, −3.21742163670067792422123583075, −2.00999573340480649906420268416, −1.08745903542785869878728096240, 1.61182540129062827467796848433, 2.72355703862033232306605000369, 3.67847504842378959872122961915, 5.07691916610455230577308034798, 5.77860193527243448943392411507, 6.72787716862398733030232285009, 7.02335850446064414888321470893, 8.587457956544430959553703294786, 9.209217621081228739350116468357, 10.17827977358227376119849779640

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