Properties

Label 2-882-441.142-c1-0-49
Degree $2$
Conductor $882$
Sign $-0.923 + 0.383i$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.365 − 0.930i)2-s + (−1.08 − 1.35i)3-s + (−0.733 − 0.680i)4-s + (2.95 − 1.42i)5-s + (−1.65 + 0.513i)6-s + (0.700 − 2.55i)7-s + (−0.900 + 0.433i)8-s + (−0.657 + 2.92i)9-s + (−0.245 − 3.27i)10-s + (0.861 + 1.08i)11-s + (−0.126 + 1.72i)12-s + (1.59 − 4.06i)13-s + (−2.11 − 1.58i)14-s + (−5.12 − 2.45i)15-s + (0.0747 + 0.997i)16-s + (4.39 − 4.07i)17-s + ⋯
L(s)  = 1  + (0.258 − 0.658i)2-s + (−0.624 − 0.780i)3-s + (−0.366 − 0.340i)4-s + (1.32 − 0.637i)5-s + (−0.675 + 0.209i)6-s + (0.264 − 0.964i)7-s + (−0.318 + 0.153i)8-s + (−0.219 + 0.975i)9-s + (−0.0775 − 1.03i)10-s + (0.259 + 0.325i)11-s + (−0.0365 + 0.498i)12-s + (0.442 − 1.12i)13-s + (−0.566 − 0.423i)14-s + (−1.32 − 0.634i)15-s + (0.0186 + 0.249i)16-s + (1.06 − 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.923 + 0.383i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.923 + 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.354279 - 1.77696i\)
\(L(\frac12)\) \(\approx\) \(0.354279 - 1.77696i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.365 + 0.930i)T \)
3 \( 1 + (1.08 + 1.35i)T \)
7 \( 1 + (-0.700 + 2.55i)T \)
good5 \( 1 + (-2.95 + 1.42i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-0.861 - 1.08i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.59 + 4.06i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-4.39 + 4.07i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-0.180 + 0.313i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.15 - 5.04i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (5.69 + 1.75i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (2.47 - 4.28i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.10 - 1.57i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-5.22 - 3.56i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (1.66 - 1.13i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-0.897 + 2.28i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-1.17 + 0.360i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (11.6 - 7.92i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (8.89 - 8.25i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (4.92 - 8.53i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.84 + 8.07i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (0.595 + 0.0897i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-5.35 - 9.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.15 - 2.95i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (3.95 + 10.0i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-1.80 + 3.13i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.936301869415397569068941612597, −9.228001002360282804607407364087, −7.934217167305296550430739370561, −7.23515646857507316979500349697, −5.91414720860605896555801511014, −5.50006359852316262141066050640, −4.55976227358099475053586007271, −3.07198064897678496880237964220, −1.65862760216231837405974468663, −0.954610847586712565705678231067, 1.92913223068342942850076196944, 3.35132984732209049671479197605, 4.45213161407150709159264255926, 5.62047900712512347887572699973, 6.00761088680424927127580529835, 6.58055278318533935303086867246, 7.992072167107872596544301624560, 9.248527108568794141632550396966, 9.333030910580079770462858105590, 10.51706395153443983469488894633

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