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Properties

Label 2-882-7.5-c2-0-28
Degree $2$
Conductor $882$
Sign $-0.832 + 0.553i$
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  + (−0.707 + 1.22i)2-s + (−0.999 − 1.73i)4-s + (−1.24 − 0.717i)5-s + 2.82·8-s + (1.75 − 1.01i)10-s + (3 + 5.19i)11-s − 21.3i·13-s + (−2.00 + 3.46i)16-s + (−7.75 + 4.47i)17-s + (6.25 + 3.61i)19-s + 2.86i·20-s − 8.48·22-s + (−18.7 + 32.4i)23-s + (−11.4 − 19.8i)25-s + (26.1 + 15.0i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.248 − 0.143i)5-s + 0.353·8-s + (0.175 − 0.101i)10-s + (0.272 + 0.472i)11-s − 1.64i·13-s + (−0.125 + 0.216i)16-s + (−0.456 + 0.263i)17-s + (0.329 + 0.190i)19-s + 0.143i·20-s − 0.385·22-s + (−0.814 + 1.41i)23-s + (−0.458 − 0.794i)25-s + (1.00 + 0.580i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.832 + 0.553i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ -0.832 + 0.553i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.05901159135\)
\(L(\frac12)\) \(\approx\) \(0.05901159135\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 1.22i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (1.24 + 0.717i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 21.3iT - 169T^{2} \)
17 \( 1 + (7.75 - 4.47i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.25 - 3.61i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (18.7 - 32.4i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 33.9T + 841T^{2} \)
31 \( 1 + (38.2 - 22.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-13.9 + 24.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 54.8iT - 1.68e3T^{2} \)
43 \( 1 + 1.48T + 1.84e3T^{2} \)
47 \( 1 + (37.2 + 21.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (42.7 + 74.0i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (35.6 - 20.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-1.02 - 0.594i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (2.19 + 3.80i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 137.T + 5.04e3T^{2} \)
73 \( 1 + (68.3 - 39.4i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (49.1 - 85.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 110. iT - 6.88e3T^{2} \)
89 \( 1 + (18 + 10.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 10.9iT - 9.40e3T^{2} \)
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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

−9.670536499240672413658208951172, −8.532229649900178369809488404932, −7.941133993929042227991850861380, −7.18616321528422551116502272142, −6.11293664785474202302029136569, −5.36049545636351860652254495697, −4.30919067929931474074369187245, −3.14249044424638942532684762080, −1.53271228902145000960551113863, −0.02201118099561768348875527335, 1.58078845379521973355521371910, 2.72173509695793278100634883192, 3.93609189687439819153110084559, 4.64321621520816492070001369899, 6.09254104810557329419197520074, 6.93475676479372237037378056802, 7.86062915008407142677930644323, 8.884535333455808068639195999675, 9.314367653517972946581414185584, 10.34547607635871603852169512669

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