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Properties

Label 2-882-21.2-c2-0-15
Degree $2$
Conductor $882$
Sign $0.230 - 0.973i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.230 - 0.973i$
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: \((2,\ 882,\ (\ :1),\ 0.230 - 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.207193161\)
\(L(\frac12)\) \(\approx\) \(3.207193161\)
\(L(2)\) 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

−10.22554878442995209975743911181, −9.217446096267044133844501124693, −8.368240567566266302688356242376, −7.44949356825359782983971228032, −6.27337104527092336052402603005, −6.08181511508364773755274582642, −4.84605870550086123496910510567, −3.79721280219042121146355296154, −2.79395526292825170200389709741, −1.47286277187066860060703507308, 0.931166697341004721690758423992, 2.14294275313713335684140583584, 3.27282942724270726807117393762, 4.47629481550908687623032957112, 5.19630773858958433967285376276, 6.25700323548979338412919310877, 6.84575335614861828177047593146, 8.217437977723435033666026030679, 9.119793546165481239034365177977, 9.688058639174488425882639635897

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