Properties

Label 2-882-21.11-c2-0-5
Degree $2$
Conductor $882$
Sign $0.516 - 0.856i$
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 + (−3.46 + 2i)5-s − 2.82i·8-s + (−2.82 + 4.89i)10-s + (2.44 + 1.41i)11-s − 12.7·13-s + (−2.00 − 3.46i)16-s + (3.46 + 2i)17-s + (11.3 + 19.5i)19-s + 7.99i·20-s + 4·22-s + (31.8 − 18.3i)23-s + (−4.50 + 7.79i)25-s + (−15.5 + 9i)26-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.692 + 0.400i)5-s − 0.353i·8-s + (−0.282 + 0.489i)10-s + (0.222 + 0.128i)11-s − 0.979·13-s + (−0.125 − 0.216i)16-s + (0.203 + 0.117i)17-s + (0.595 + 1.03i)19-s + 0.399i·20-s + 0.181·22-s + (1.38 − 0.799i)23-s + (−0.180 + 0.311i)25-s + (−0.599 + 0.346i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.516 - 0.856i$
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: \((2,\ 882,\ (\ :1),\ 0.516 - 0.856i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.814688972\)
\(L(\frac12)\) \(\approx\) \(1.814688972\)
\(L(2)\) 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 \( 1 + (3.46 - 2i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.44 - 1.41i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 12.7T + 169T^{2} \)
17 \( 1 + (-3.46 - 2i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-11.3 - 19.5i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-31.8 + 18.3i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 32.5iT - 841T^{2} \)
31 \( 1 + (25.4 - 44.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-16 - 27.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 38iT - 1.68e3T^{2} \)
43 \( 1 - 20T + 1.84e3T^{2} \)
47 \( 1 + (-17.3 + 10i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (82.0 + 47.3i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-3.46 - 2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-41.7 - 72.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-24 + 41.5i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 76.3iT - 5.04e3T^{2} \)
73 \( 1 + (60.1 - 104. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-74 - 128. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 80iT - 6.88e3T^{2} \)
89 \( 1 + (-91.7 + 53i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 154.T + 9.40e3T^{2} \)
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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

−10.23566717888982491829405125711, −9.414166597400140937723540675014, −8.335562962466227174217705607821, −7.30373798104650371253342672332, −6.76806511574085737315114774793, −5.47943906281718071687014980754, −4.70896570098605358507546268521, −3.59277670969458377539430381065, −2.84720030382197241611708606762, −1.34667483154596513131380278529, 0.50733187373407850712237596916, 2.38984525078270648995136636718, 3.54934514290746481322033780878, 4.51704006121080862112696233591, 5.25338046933106047452307510064, 6.27858765065281119359703061958, 7.53370464747188420454586406450, 7.62771987949498439451612709576, 9.027109521958457304156800065589, 9.551092178462214810476538812184

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