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Properties

Label 2-882-7.4-c3-0-4
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 + (7.61 + 13.1i)5-s − 7.99·8-s + (−15.2 + 26.3i)10-s + (1 − 1.73i)11-s − 30.4·13-s + (−8 − 13.8i)16-s + (−22.8 + 39.5i)17-s + (−76.1 − 131. i)19-s − 60.9·20-s + 3.99·22-s + (15 + 25.9i)23-s + (−53.5 + 92.6i)25-s + (−30.4 − 52.7i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.681 + 1.17i)5-s − 0.353·8-s + (−0.481 + 0.834i)10-s + (0.0274 − 0.0474i)11-s − 0.649·13-s + (−0.125 − 0.216i)16-s + (−0.325 + 0.564i)17-s + (−0.919 − 1.59i)19-s − 0.681·20-s + 0.0387·22-s + (0.135 + 0.235i)23-s + (−0.427 + 0.741i)25-s + (−0.229 − 0.397i)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

\(L(2)\) \(\approx\) \(0.4720006071\)
\(L(\frac12)\) \(\approx\) \(0.4720006071\)
\(L(\frac{5}{2})\) 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 \( 1 + (-7.61 - 13.1i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 30.4T + 2.19e3T^{2} \)
17 \( 1 + (22.8 - 39.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (76.1 + 131. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-15 - 25.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 212T + 2.43e4T^{2} \)
31 \( 1 + (106. - 184. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (123 + 213. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 319.T + 6.89e4T^{2} \)
43 \( 1 + 284T + 7.95e4T^{2} \)
47 \( 1 + (-30.4 - 52.7i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-274 + 474. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (335. - 580. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-258. - 448. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (326 - 564. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 770T + 3.57e5T^{2} \)
73 \( 1 + (-487. + 844. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (236 + 408. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 182.T + 5.71e5T^{2} \)
89 \( 1 + (-357. - 619. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 304.T + 9.12e5T^{2} \)
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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

−10.46265590918597910935047796627, −9.352691454292576040557861113065, −8.707729363701688104433105309085, −7.36924677766897018023277649476, −6.94630566046592108623494841632, −6.09727640575604641066788536107, −5.24103933116025458849747377254, −4.11718552889467539479514561367, −2.96446988163331322804508191210, −2.05809951818547145869280592465, 0.097943365115504219589888462407, 1.47819903659126465333033178469, 2.29952039664048199560811616951, 3.76579400741589641895325400437, 4.69056241883356297286603984709, 5.46693889708408026343265023005, 6.25346904542678486950426535266, 7.58094688845543913001788172086, 8.562493767231141675149049262198, 9.371829940585160894438805782440

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