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Properties

Label 2-882-7.4-c3-0-8
Degree $2$
Conductor $882$
Sign $-0.749 - 0.661i$
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 + (9.89 + 17.1i)5-s + 7.99·8-s + (19.7 − 34.2i)10-s + (−7 + 12.1i)11-s − 50.9·13-s + (−8 − 13.8i)16-s + (−0.707 + 1.22i)17-s + (−0.707 − 1.22i)19-s − 79.1·20-s + 28·22-s + (70 + 121. i)23-s + (−133. + 231. i)25-s + (50.9 + 88.1i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.885 + 1.53i)5-s + 0.353·8-s + (0.626 − 1.08i)10-s + (−0.191 + 0.332i)11-s − 1.08·13-s + (−0.125 − 0.216i)16-s + (−0.0100 + 0.0174i)17-s + (−0.00853 − 0.0147i)19-s − 0.885·20-s + 0.271·22-s + (0.634 + 1.09i)23-s + (−1.06 + 1.84i)25-s + (0.384 + 0.665i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.749 - 0.661i$
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: \((2,\ 882,\ (\ :3/2),\ -0.749 - 0.661i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.020718232\)
\(L(\frac12)\) \(\approx\) \(1.020718232\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.89 - 17.1i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (7 - 12.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 50.9T + 2.19e3T^{2} \)
17 \( 1 + (0.707 - 1.22i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (0.707 + 1.22i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-70 - 121. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 286T + 2.43e4T^{2} \)
31 \( 1 + (46.6 - 80.8i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-19 - 32.9i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 125.T + 6.89e4T^{2} \)
43 \( 1 + 34T + 7.95e4T^{2} \)
47 \( 1 + (261. + 453. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (37 - 64.0i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (217. - 375. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-7.07 - 12.2i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (342 - 592. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 588T + 3.57e5T^{2} \)
73 \( 1 + (135. - 233. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (610 + 1.05e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 422.T + 5.71e5T^{2} \)
89 \( 1 + (309. + 535. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.48e3T + 9.12e5T^{2} \)
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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

−10.08919139480614310884990104658, −9.634894127309001858393137118853, −8.539653947480992734243581680193, −7.30620937459761632531189523164, −6.92108630286864263786261003014, −5.77116766179891848188545536818, −4.71090521018591599807868959071, −3.23853276184909593166650410231, −2.63256124699077910833890179496, −1.60287146094368946268451235143, 0.29449209987734126769555513327, 1.34860642810361518569607493128, 2.63019114901879984600017364355, 4.55201335185309280783834332753, 4.98211292346694978585814043897, 5.92602239707001080931848776923, 6.76550200499265816108819115919, 7.988719808671418234806438944536, 8.556022804426205005955897169437, 9.376397320821657565383681027425

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