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Properties

Label 2-882-7.4-c3-0-15
Degree $2$
Conductor $882$
Sign $-0.991 + 0.126i$
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 + (4.5 + 7.79i)5-s − 7.99·8-s + (−9 + 15.5i)10-s + (−28.5 + 49.3i)11-s + 70·13-s + (−8 − 13.8i)16-s + (−25.5 + 44.1i)17-s + (2.5 + 4.33i)19-s − 36·20-s − 114·22-s + (34.5 + 59.7i)23-s + (22 − 38.1i)25-s + (70 + 121. i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.402 + 0.697i)5-s − 0.353·8-s + (−0.284 + 0.492i)10-s + (−0.781 + 1.35i)11-s + 1.49·13-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + (0.0301 + 0.0522i)19-s − 0.402·20-s − 1.10·22-s + (0.312 + 0.541i)23-s + (0.175 − 0.304i)25-s + (0.528 + 0.914i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
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: \((2,\ 882,\ (\ :3/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.869677161\)
\(L(\frac12)\) \(\approx\) \(1.869677161\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.5 - 7.79i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (28.5 - 49.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 70T + 2.19e3T^{2} \)
17 \( 1 + (25.5 - 44.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-34.5 - 59.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 114T + 2.43e4T^{2} \)
31 \( 1 + (-11.5 + 19.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-126.5 - 219. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 42T + 6.89e4T^{2} \)
43 \( 1 + 124T + 7.95e4T^{2} \)
47 \( 1 + (100.5 + 174. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (196.5 - 340. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (109.5 - 189. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (354.5 + 614. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (209.5 - 362. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 96T + 3.57e5T^{2} \)
73 \( 1 + (156.5 - 271. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (230.5 + 399. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 588T + 5.71e5T^{2} \)
89 \( 1 + (-508.5 - 880. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.83e3T + 9.12e5T^{2} \)
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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

−10.22248605004855437213955097624, −9.315033098008783370951175670099, −8.326203716411099886365180275606, −7.53449368960982168236250686623, −6.64156583006287359659093383740, −6.01412956470449521866673363870, −4.98074150014480016222205894316, −3.98379792382082876720006959059, −2.89158597003839989872205287195, −1.67020384035870446609960761170, 0.42438150281191247574806351189, 1.44475935147327942645373198978, 2.80326950899123060341505491075, 3.71313759502123318246440156023, 4.88932624879956249380394581299, 5.66057308737277947206967451655, 6.39115237548035909812671913911, 7.81354504870175813919754011897, 8.785101259799875944939989603785, 9.147409219671262156311145214390

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