Properties

Label 2-882-7.4-c3-0-32
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 + (−4 − 6.92i)5-s + 7.99·8-s + (−7.99 + 13.8i)10-s + (20 − 34.6i)11-s − 4·13-s + (−8 − 13.8i)16-s + (42 − 72.7i)17-s + (74 + 128. i)19-s + 31.9·20-s − 80·22-s + (42 + 72.7i)23-s + (30.5 − 52.8i)25-s + (4 + 6.92i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.357 − 0.619i)5-s + 0.353·8-s + (−0.252 + 0.438i)10-s + (0.548 − 0.949i)11-s − 0.0853·13-s + (−0.125 − 0.216i)16-s + (0.599 − 1.03i)17-s + (0.893 + 1.54i)19-s + 0.357·20-s − 0.775·22-s + (0.380 + 0.659i)23-s + (0.244 − 0.422i)25-s + (0.0301 + 0.0522i)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

\(L(2)\) \(\approx\) \(1.505015403\)
\(L(\frac12)\) \(\approx\) \(1.505015403\)
\(L(\frac{5}{2})\) 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 \( 1 + (4 + 6.92i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-20 + 34.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 4T + 2.19e3T^{2} \)
17 \( 1 + (-42 + 72.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-74 - 128. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-42 - 72.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 58T + 2.43e4T^{2} \)
31 \( 1 + (68 - 117. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-111 - 192. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 420T + 6.89e4T^{2} \)
43 \( 1 + 164T + 7.95e4T^{2} \)
47 \( 1 + (244 + 422. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-239 + 413. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (274 - 474. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-346 - 599. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-454 + 786. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 524T + 3.57e5T^{2} \)
73 \( 1 + (-220 + 381. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (608 + 1.05e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 684T + 5.71e5T^{2} \)
89 \( 1 + (302 + 523. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 832T + 9.12e5T^{2} \)
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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

−9.528140775371864134018027101228, −8.712882560227662252369905312791, −8.005443474786458595116267818284, −7.16423219140379823302192824912, −5.85287937243293545413355382586, −4.98498115317466576903732732524, −3.79315530065622872314601504341, −3.06494867622510365688247579511, −1.47543416799411403831628700372, −0.57707210975312479493743956858, 1.00421005397588998533956616212, 2.47656822013302420966938165988, 3.77550680248969001489138332926, 4.75992817625116901059784829602, 5.82198346397583932902458569575, 6.84522350296149077361290414139, 7.32867786215593760669409647592, 8.179525701153965263108620433752, 9.306199235491868336321694692864, 9.714879919780118228268093871157

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