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Properties

Label 2-882-441.142-c1-0-23
Degree $2$
Conductor $882$
Sign $0.262 + 0.964i$
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 + (−0.987 + 1.42i)3-s + (−0.733 − 0.680i)4-s + (−3.29 + 1.58i)5-s + (0.964 + 1.43i)6-s + (−2.61 + 0.431i)7-s + (−0.900 + 0.433i)8-s + (−1.05 − 2.80i)9-s + (0.273 + 3.64i)10-s + (0.844 + 1.05i)11-s + (1.69 − 0.371i)12-s + (−0.995 + 2.53i)13-s + (−0.551 + 2.58i)14-s + (0.993 − 6.24i)15-s + (0.0747 + 0.997i)16-s + (2.70 − 2.50i)17-s + ⋯
L(s)  = 1  + (0.258 − 0.658i)2-s + (−0.569 + 0.821i)3-s + (−0.366 − 0.340i)4-s + (−1.47 + 0.708i)5-s + (0.393 + 0.587i)6-s + (−0.986 + 0.163i)7-s + (−0.318 + 0.153i)8-s + (−0.350 − 0.936i)9-s + (0.0863 + 1.15i)10-s + (0.254 + 0.319i)11-s + (0.488 − 0.107i)12-s + (−0.276 + 0.703i)13-s + (−0.147 + 0.691i)14-s + (0.256 − 1.61i)15-s + (0.0186 + 0.249i)16-s + (0.655 − 0.607i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.262 + 0.964i$
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: \((2,\ 882,\ (\ :1/2),\ 0.262 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.428823 - 0.327789i\)
\(L(\frac12)\) \(\approx\) \(0.428823 - 0.327789i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.987 - 1.42i)T \)
7 \( 1 + (2.61 - 0.431i)T \)
good5 \( 1 + (3.29 - 1.58i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (-0.844 - 1.05i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.995 - 2.53i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-2.70 + 2.50i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-1.45 + 2.51i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.85 + 8.12i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-5.84 - 1.80i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (5.31 - 9.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.59 + 1.10i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (7.13 + 4.86i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (-5.63 + 3.83i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-0.213 + 0.544i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (1.97 - 0.609i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-9.71 + 6.62i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (1.15 - 1.07i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (2.56 - 4.43i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.44 + 6.35i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (0.900 + 0.135i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (0.821 + 1.42i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.03 + 7.72i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-1.93 - 4.93i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-1.06 + 1.83i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.34874047357685227872196709436, −9.264770186664870064069120823523, −8.620473574733856786372339623218, −7.00998857609644512208023567404, −6.72009981787097324623635873491, −5.24019241691314804173189085363, −4.39299150237001009392947453750, −3.52220507092381282042942405699, −2.86868142133396474200678005409, −0.34932174517328319531108857832, 0.943645736496364123809729388712, 3.22030433377268298724591129084, 4.03384906101948622193903946569, 5.27494981067544512165281363617, 5.95575487834066937141769445883, 6.99903665854591471105853533559, 7.77208771283118217006090587338, 8.132281482707874069888009908074, 9.277494688191202848340498317420, 10.33747386681280621585092545878

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