Properties

Label 2-882-7.4-c3-0-6
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 + (9 + 15.5i)5-s − 7.99·8-s + (−18 + 31.1i)10-s + (−36 + 62.3i)11-s − 34·13-s + (−8 − 13.8i)16-s + (3 − 5.19i)17-s + (−46 − 79.6i)19-s − 72·20-s − 144·22-s + (−90 − 155. i)23-s + (−99.5 + 172. i)25-s + (−34 − 58.8i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.804 + 1.39i)5-s − 0.353·8-s + (−0.569 + 0.985i)10-s + (−0.986 + 1.70i)11-s − 0.725·13-s + (−0.125 − 0.216i)16-s + (0.0428 − 0.0741i)17-s + (−0.555 − 0.962i)19-s − 0.804·20-s − 1.39·22-s + (−0.815 − 1.41i)23-s + (−0.796 + 1.37i)25-s + (−0.256 − 0.444i)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

\(L(2)\) \(\approx\) \(0.9578196484\)
\(L(\frac12)\) \(\approx\) \(0.9578196484\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9 - 15.5i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (36 - 62.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 34T + 2.19e3T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (46 + 79.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (90 + 155. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 114T + 2.43e4T^{2} \)
31 \( 1 + (28 - 48.4i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-17 - 29.4i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 6T + 6.89e4T^{2} \)
43 \( 1 - 164T + 7.95e4T^{2} \)
47 \( 1 + (-84 - 145. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-327 + 566. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (246 - 426. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-125 - 216. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-62 + 107. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 36T + 3.57e5T^{2} \)
73 \( 1 + (505 - 874. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (28 + 48.4i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 228T + 5.71e5T^{2} \)
89 \( 1 + (-195 - 337. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 70T + 9.12e5T^{2} \)
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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

−10.17682544642988609784279784092, −9.752242672255233194586873788912, −8.505714398240621808622127694839, −7.36705970885382882934481207078, −6.97043644373100683648991830127, −6.17035580961689196971398604650, −5.09195024477580779313941494511, −4.29705270524887573387812515617, −2.66849887594868981598694219678, −2.30402539244221508885544116223, 0.20559362910245983440748026613, 1.33184794008812499500985889060, 2.42791625958768324447019374584, 3.64810735599808363173423859172, 4.76860338228248145037761451940, 5.64520922713815368824224805549, 5.97738217449239003240660322160, 7.76184574712418731343228700030, 8.486706441210684143969890904634, 9.257968920093157126934108415690

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