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Properties

Label 2-882-7.4-c3-0-45
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (1 + 1.73i)5-s − 7.99·8-s + (−1.99 + 3.46i)10-s + (−4 + 6.92i)11-s − 42·13-s + (−8 − 13.8i)16-s + (−1 + 1.73i)17-s + (62 + 107. i)19-s − 7.99·20-s − 15.9·22-s + (38 + 65.8i)23-s + (60.5 − 104. i)25-s + (−42 − 72.7i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.0894 + 0.154i)5-s − 0.353·8-s + (−0.0632 + 0.109i)10-s + (−0.109 + 0.189i)11-s − 0.896·13-s + (−0.125 − 0.216i)16-s + (−0.0142 + 0.0247i)17-s + (0.748 + 1.29i)19-s − 0.0894·20-s − 0.155·22-s + (0.344 + 0.596i)23-s + (0.483 − 0.838i)25-s + (−0.316 − 0.548i)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: \(1\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1 - 1.73i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (4 - 6.92i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 42T + 2.19e3T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-62 - 107. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-38 - 65.8i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 254T + 2.43e4T^{2} \)
31 \( 1 + (-36 + 62.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (199 + 344. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 462T + 6.89e4T^{2} \)
43 \( 1 - 212T + 7.95e4T^{2} \)
47 \( 1 + (132 + 228. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (81 - 140. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (386 - 668. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (15 + 25.9i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-382 + 661. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 236T + 3.57e5T^{2} \)
73 \( 1 + (209 - 361. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (276 + 478. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.03e3T + 5.71e5T^{2} \)
89 \( 1 + (-15 - 25.9i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.19e3T + 9.12e5T^{2} \)
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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

−9.492125427186425551819928777797, −8.516685608785929237765059703184, −7.54268344911022424217676066981, −7.06702745227632497344772886662, −5.86329474625193205020151169480, −5.25050797376140655598605551675, −4.13769172186625743973252755457, −3.15316868092504720130332980679, −1.83686934965998388344058772584, 0, 1.36187223808993675589383215702, 2.62852374635398019471148225153, 3.48804790107523701954512735509, 4.86326167096767896252400760059, 5.23424739091545880754349025379, 6.58568620512480671500002888223, 7.36567973164797967155498064495, 8.532536226072913250494392903732, 9.348059075884290558662044359225

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