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Properties

Label 2-882-9.4-c1-0-0
Degree $2$
Conductor $882$
Sign $-0.704 + 0.709i$
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  + (0.5 + 0.866i)2-s + (−1.73 + 0.0789i)3-s + (−0.499 + 0.866i)4-s + (−0.296 + 0.514i)5-s + (−0.933 − 1.45i)6-s − 0.999·8-s + (2.98 − 0.273i)9-s − 0.593·10-s + (0.296 + 0.514i)11-s + (0.796 − 1.53i)12-s + (−1.25 + 2.17i)13-s + (0.472 − 0.912i)15-s + (−0.5 − 0.866i)16-s − 2.92·17-s + (1.73 + 2.45i)18-s − 5.38·19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.998 + 0.0455i)3-s + (−0.249 + 0.433i)4-s + (−0.132 + 0.229i)5-s + (−0.381 − 0.595i)6-s − 0.353·8-s + (0.995 − 0.0910i)9-s − 0.187·10-s + (0.0894 + 0.154i)11-s + (0.230 − 0.443i)12-s + (−0.348 + 0.603i)13-s + (0.122 − 0.235i)15-s + (−0.125 − 0.216i)16-s − 0.708·17-s + (0.407 + 0.577i)18-s − 1.23·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.704 + 0.709i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.704 + 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0882236 - 0.211827i\)
\(L(\frac12)\) \(\approx\) \(0.0882236 - 0.211827i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (1.73 - 0.0789i)T \)
7 \( 1 \)
good5 \( 1 + (0.296 - 0.514i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.296 - 0.514i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.25 - 2.17i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.92T + 17T^{2} \)
19 \( 1 + 5.38T + 19T^{2} \)
23 \( 1 + (2.23 - 3.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.09 + 5.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.93 + 6.81i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (0.136 - 0.236i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.58 + 9.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.08 + 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.05T + 53T^{2} \)
59 \( 1 + (4.32 - 7.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.32 - 5.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.956 + 1.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + 7.91T + 73T^{2} \)
79 \( 1 + (-4.62 - 8.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.85 - 6.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 + (-5.86 - 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.74459978278274174410722530909, −9.860086411590712423955127214763, −8.991203367486727195847385483440, −7.86663287456010923101690704479, −6.99169415466287488997164681995, −6.39510371591658115479812091303, −5.51049573886526661068200717327, −4.53850132453205447425762157160, −3.81996255194929187714955641137, −2.04576080051073008084162770614, 0.10828789570721755928625206493, 1.64015263602905558585596955921, 3.07370583650086279752459969745, 4.50191822827864400308483692247, 4.85392502338510091964892379810, 6.14689041370445197938997457968, 6.65385406980951652173257506389, 7.976418650798262477420662657213, 8.877483311180581816288067855838, 9.967169208886161346820599256340

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