Properties

Label 2-882-21.2-c2-0-21
Degree $2$
Conductor $882$
Sign $-0.645 + 0.763i$
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 + (14.6 − 8.48i)11-s − 8·13-s + (−2.00 + 3.46i)16-s + (11.0 − 6.36i)17-s + (−8 + 13.8i)19-s − 8.48i·20-s − 24·22-s + (14.6 + 8.48i)23-s + (−3.5 − 6.06i)25-s + (9.79 + 5.65i)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 + (1.33 − 0.771i)11-s − 0.615·13-s + (−0.125 + 0.216i)16-s + (0.648 − 0.374i)17-s + (−0.421 + 0.729i)19-s − 0.424i·20-s − 1.09·22-s + (0.638 + 0.368i)23-s + (−0.140 − 0.242i)25-s + (0.376 + 0.217i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.645 + 0.763i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ -0.645 + 0.763i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8643851527\)
\(L(\frac12)\) \(\approx\) \(0.8643851527\)
\(L(2)\) 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 \( 1 + (-14.6 + 8.48i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 8T + 169T^{2} \)
17 \( 1 + (-11.0 + 6.36i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (8 - 13.8i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-14.6 - 8.48i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 4.24iT - 841T^{2} \)
31 \( 1 + (-22 - 38.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-17 + 29.4i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 46.6iT - 1.68e3T^{2} \)
43 \( 1 + 40T + 1.84e3T^{2} \)
47 \( 1 + (73.4 + 42.4i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-33.0 + 19.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (29.3 - 16.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-25 + 43.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 50.9iT - 5.04e3T^{2} \)
73 \( 1 + (8 + 13.8i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-38 + 65.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 118. iT - 6.88e3T^{2} \)
89 \( 1 + (-11.0 - 6.36i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 176T + 9.40e3T^{2} \)
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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

−9.575975950268339628134603841893, −8.761408250986675665963867266210, −8.171580806826909560975255319066, −7.23165564165952919872077076216, −6.36437199675942058731137304594, −5.11393037739498814138157043515, −3.96506829180250120192089407213, −3.18961281053659149247840428680, −1.59101870576369595937035838314, −0.39739311974999042794728534271, 1.23593945395795087787291779759, 2.70082817516649637689024406973, 3.99551639365702219500021050689, 4.88923233735178366573834220899, 6.30094403737312104757359129181, 6.88685179351601789470443518664, 7.70766503201771374998511878492, 8.454187802152264854383250394307, 9.543704094549151842834607024469, 9.928113462314503542509869471272

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