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Properties

Label 2-882-441.142-c1-0-1
Degree $2$
Conductor $882$
Sign $-0.397 + 0.917i$
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 + (1.29 + 1.15i)3-s + (−0.733 − 0.680i)4-s + (−1.94 + 0.938i)5-s + (−1.54 + 0.785i)6-s + (−2.64 − 0.140i)7-s + (0.900 − 0.433i)8-s + (0.353 + 2.97i)9-s + (−0.161 − 2.15i)10-s + (−0.448 − 0.562i)11-s + (−0.166 − 1.72i)12-s + (−1.22 + 3.13i)13-s + (1.09 − 2.40i)14-s + (−3.60 − 1.02i)15-s + (0.0747 + 0.997i)16-s + (0.852 − 0.791i)17-s + ⋯
L(s)  = 1  + (−0.258 + 0.658i)2-s + (0.747 + 0.664i)3-s + (−0.366 − 0.340i)4-s + (−0.871 + 0.419i)5-s + (−0.630 + 0.320i)6-s + (−0.998 − 0.0531i)7-s + (0.318 − 0.153i)8-s + (0.117 + 0.993i)9-s + (−0.0511 − 0.682i)10-s + (−0.135 − 0.169i)11-s + (−0.0481 − 0.497i)12-s + (−0.340 + 0.868i)13-s + (0.292 − 0.643i)14-s + (−0.930 − 0.265i)15-s + (0.0186 + 0.249i)16-s + (0.206 − 0.191i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.397 + 0.917i$
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: \((2,\ 882,\ (\ :1/2),\ -0.397 + 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.153534 - 0.233773i\)
\(L(\frac12)\) \(\approx\) \(0.153534 - 0.233773i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 - 1.15i)T \)
7 \( 1 + (2.64 + 0.140i)T \)
good5 \( 1 + (1.94 - 0.938i)T + (3.11 - 3.90i)T^{2} \)
11 \( 1 + (0.448 + 0.562i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.22 - 3.13i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-0.852 + 0.791i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (0.379 - 0.656i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.76 + 7.72i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (9.23 + 2.84i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (-2.90 + 5.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.66 + 1.12i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (5.47 + 3.73i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (3.14 - 2.14i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (-3.90 + 9.94i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (6.47 - 1.99i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (4.04 - 2.75i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (4.32 - 4.00i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (5.74 - 9.94i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.26 - 14.2i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-9.77 - 1.47i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-2.73 - 4.74i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.62 - 9.23i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-3.63 - 9.27i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (7.86 - 13.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.44474580785685230565413143716, −9.704509142322674408425561214737, −9.015628027670871590605050265190, −8.208661615269773391813237125869, −7.34353146625401799812746762169, −6.70875206956153089996131401250, −5.49805760837106978226754641642, −4.27425171674884619901053602067, −3.63746821478686060543226187900, −2.44812377143134629487051732380, 0.12962550181479752504632181189, 1.64981823996310066372148230588, 3.19419740376159885536686425202, 3.49267267416525520463926815701, 4.92032346454067006010400675705, 6.22791522995234309720797482430, 7.45552994102372208739337588035, 7.79091835614907925391980252158, 8.847404722183717823139850547492, 9.431724254643933230901761977622

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