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Properties

Label 2-882-7.4-c3-0-3
Degree $2$
Conductor $882$
Sign $0.386 - 0.922i$
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 − 12.1i)5-s + 7.99·8-s + (−14 + 24.2i)10-s + (−14 + 24.2i)11-s + 18·13-s + (−8 − 13.8i)16-s + (37 − 64.0i)17-s + (−40 − 69.2i)19-s + 56·20-s + 56·22-s + (−56 − 96.9i)23-s + (−35.5 + 61.4i)25-s + (−18 − 31.1i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.626 − 1.08i)5-s + 0.353·8-s + (−0.442 + 0.766i)10-s + (−0.383 + 0.664i)11-s + 0.384·13-s + (−0.125 − 0.216i)16-s + (0.527 − 0.914i)17-s + (−0.482 − 0.836i)19-s + 0.626·20-s + 0.542·22-s + (−0.507 − 0.879i)23-s + (−0.284 + 0.491i)25-s + (−0.135 − 0.235i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
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: \((2,\ 882,\ (\ :3/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2203438958\)
\(L(\frac12)\) \(\approx\) \(0.2203438958\)
\(L(\frac{5}{2})\) 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 \( 1 + (7 + 12.1i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (14 - 24.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 18T + 2.19e3T^{2} \)
17 \( 1 + (-37 + 64.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (40 + 69.2i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (56 + 96.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 190T + 2.43e4T^{2} \)
31 \( 1 + (36 - 62.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-173 - 299. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 162T + 6.89e4T^{2} \)
43 \( 1 + 412T + 7.95e4T^{2} \)
47 \( 1 + (-12 - 20.7i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-159 + 275. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (100 - 173. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-99 - 171. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-358 + 620. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 392T + 3.57e5T^{2} \)
73 \( 1 + (269 - 465. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (120 + 207. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.07e3T + 5.71e5T^{2} \)
89 \( 1 + (-405 - 701. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.35e3T + 9.12e5T^{2} \)
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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

−9.861393497598786178970393283272, −9.071233578554283400808104945153, −8.349015558766799485354970800141, −7.66167613167549087394831792121, −6.61957134208547286845542164106, −5.12879999039638708641614389843, −4.57818685151988653673035464103, −3.49665243315996359222679105096, −2.23352628283354131143899657181, −0.953689675696556563660130204592, 0.07849625838598364591452362339, 1.78455760980068551552377119461, 3.32963772638645967457239571306, 3.98858041896671734672985763494, 5.58437660317652174596301117371, 6.12487352319705988304288386826, 7.18290496778931765504583433834, 7.84602910106019647461726148471, 8.487162287820145737908884077916, 9.606478394605896778823228647951

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