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Properties

Label 2-882-7.4-c3-0-48
Degree $2$
Conductor $882$
Sign $-0.749 - 0.661i$
Analytic cond. $52.0396$
Root an. cond. $7.21385$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s + (−7.94 − 13.7i)5-s + 7.99·8-s + (−15.8 + 27.5i)10-s + (28.6 − 49.7i)11-s + 5.69·13-s + (−8 − 13.8i)16-s + (−25.9 + 44.9i)17-s + (−8.10 − 14.0i)19-s + 63.5·20-s − 114.·22-s + (−106. − 184. i)23-s + (−63.8 + 110. i)25-s + (−5.69 − 9.87i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.711 − 1.23i)5-s + 0.353·8-s + (−0.502 + 0.870i)10-s + (0.786 − 1.36i)11-s + 0.121·13-s + (−0.125 − 0.216i)16-s + (−0.370 + 0.641i)17-s + (−0.0978 − 0.169i)19-s + 0.711·20-s − 1.11·22-s + (−0.967 − 1.67i)23-s + (−0.511 + 0.885i)25-s + (−0.0429 − 0.0744i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.749 - 0.661i$
Analytic conductor: \(52.0396\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ -0.749 - 0.661i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8951659136\)
\(L(\frac12)\) \(\approx\) \(0.8951659136\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (7.94 + 13.7i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-28.6 + 49.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 5.69T + 2.19e3T^{2} \)
17 \( 1 + (25.9 - 44.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (8.10 + 14.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (106. + 184. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 218.T + 2.43e4T^{2} \)
31 \( 1 + (-125. + 217. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (193. + 334. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 328.T + 6.89e4T^{2} \)
43 \( 1 + 37.5T + 7.95e4T^{2} \)
47 \( 1 + (127. + 220. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-105. + 183. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (206. - 356. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-418. - 724. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-82.7 + 143. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 465.T + 3.57e5T^{2} \)
73 \( 1 + (224. - 389. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-171. - 297. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.50e3T + 5.71e5T^{2} \)
89 \( 1 + (170. + 295. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 865.T + 9.12e5T^{2} \)
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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

−8.969840227332501784627931168995, −8.535102053132901416042190644055, −8.037737848341833286451020074022, −6.63435402554232289899969596618, −5.67161880275923541610645846802, −4.31607133646576587509335385882, −3.95542442351301626573021105019, −2.50852822176425141933881045826, −1.03569779480578654913395751210, −0.32167538334763363634013089493, 1.51788068164673556313036386908, 2.94699820172751735506954862945, 4.04116870444629167761492011433, 4.97781725839908971119899705546, 6.38979738854521593768568070889, 6.87632606527252600444542489130, 7.57786255518170502866670822049, 8.390814301433001914509035818717, 9.571729569245443143546721719027, 10.04737761955339762147626315090

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