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Properties

Label 2-882-21.11-c2-0-13
Degree $2$
Conductor $882$
Sign $0.882 + 0.469i$
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 + (3.67 − 2.12i)5-s + 2.82i·8-s + (−3 + 5.19i)10-s + (−11.0 − 6.36i)11-s + 13-s + (−2.00 − 3.46i)16-s + (14.6 + 8.48i)17-s + (11.5 + 19.9i)19-s − 8.48i·20-s + 18·22-s + (14.6 − 8.48i)23-s + (−3.5 + 6.06i)25-s + (−1.22 + 0.707i)26-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.734 − 0.424i)5-s + 0.353i·8-s + (−0.300 + 0.519i)10-s + (−1.00 − 0.578i)11-s + 0.0769·13-s + (−0.125 − 0.216i)16-s + (0.864 + 0.499i)17-s + (0.605 + 1.04i)19-s − 0.424i·20-s + 0.818·22-s + (0.638 − 0.368i)23-s + (−0.140 + 0.242i)25-s + (−0.0471 + 0.0271i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.882 + 0.469i$
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: \((2,\ 882,\ (\ :1),\ 0.882 + 0.469i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.498758783\)
\(L(\frac12)\) \(\approx\) \(1.498758783\)
\(L(2)\) 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 \( 1 + (-3.67 + 2.12i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (11.0 + 6.36i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - T + 169T^{2} \)
17 \( 1 + (-14.6 - 8.48i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-11.5 - 19.9i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-14.6 + 8.48i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 33.9iT - 841T^{2} \)
31 \( 1 + (-23.5 + 40.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-27.5 - 47.6i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 46.6iT - 1.68e3T^{2} \)
43 \( 1 - 23T + 1.84e3T^{2} \)
47 \( 1 + (3.67 - 2.12i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (44.0 + 25.4i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (73.4 + 42.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-52 - 90.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-48.5 + 84.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 97.5iT - 5.04e3T^{2} \)
73 \( 1 + (-32.5 + 56.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (56.5 + 97.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 29.6iT - 6.88e3T^{2} \)
89 \( 1 + (-117. + 67.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 104T + 9.40e3T^{2} \)
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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

−9.865719310111557435783879479226, −9.083833636110077752395692940315, −8.022087726852077189581433658717, −7.70711218587262166814207742321, −6.18254520411938844040178923367, −5.78130473143812503631309748912, −4.79059161416349287125462381075, −3.27791016337674869306805119521, −1.99480493508842046158376806345, −0.72349979929513840154668891986, 1.07883334071412908599988494406, 2.46513254758780108580677343838, 3.14950856175465587134293012830, 4.76748319442098445423032070166, 5.61323856939952557296600190676, 6.83204225245243557456870946701, 7.43905607875893766092125162119, 8.403529008296025653251961079394, 9.451068398618163698043544787643, 9.890920030417361540936865427155

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