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Properties

Label 2-882-7.4-c3-0-35
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

\(L(2)\) \(\approx\) \(1.262254232\)
\(L(\frac12)\) \(\approx\) \(1.262254232\)
\(L(\frac{5}{2})\) 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

−9.520824818266280354133375905641, −8.746484285154261989225518921380, −8.069712964044986680870957581835, −6.72590039828169247260887310317, −6.28986634520893858265906278529, −4.90822406959509140830949012500, −3.79843187422710614307505161963, −2.85382632787840991472545488910, −1.80638605263437487070091983098, −0.40372768027252831420413483416, 1.12280294265039273695632535460, 2.22051265698941903484376616446, 3.94561779717954434556179207480, 4.84168424903329939716650601645, 5.69226922168415090878288138083, 6.72557698150347431004615973683, 7.34338592388613631723495004630, 8.552416923603640979856184080166, 8.965411909394660092936402786834, 9.910075997239317237751294079883

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