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Properties

Label 2-882-7.2-c3-0-42
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: $\chi_{882} (667, \cdot )$
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

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

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