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Properties

Label 2-882-63.4-c1-0-21
Degree $2$
Conductor $882$
Sign $0.853 + 0.521i$
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 + (1.22 + 1.22i)3-s + (−0.499 − 0.866i)4-s − 3.86·5-s + (−1.67 + 0.448i)6-s + 0.999·8-s + 2.99i·9-s + (1.93 − 3.34i)10-s − 3.73·11-s + (0.448 − 1.67i)12-s + (3.34 − 5.79i)13-s + (−4.73 − 4.73i)15-s + (−0.5 + 0.866i)16-s + (2.70 − 4.69i)17-s + (−2.59 − 1.49i)18-s + (−1.48 − 2.56i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.707 + 0.707i)3-s + (−0.249 − 0.433i)4-s − 1.72·5-s + (−0.683 + 0.183i)6-s + 0.353·8-s + 0.999i·9-s + (0.610 − 1.05i)10-s − 1.12·11-s + (0.129 − 0.482i)12-s + (0.928 − 1.60i)13-s + (−1.22 − 1.22i)15-s + (−0.125 + 0.216i)16-s + (0.656 − 1.13i)17-s + (−0.612 − 0.353i)18-s + (−0.340 − 0.589i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.853 + 0.521i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.853 + 0.521i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.701102 - 0.197225i\)
\(L(\frac12)\) \(\approx\) \(0.701102 - 0.197225i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.22 - 1.22i)T \)
7 \( 1 \)
good5 \( 1 + 3.86T + 5T^{2} \)
11 \( 1 + 3.73T + 11T^{2} \)
13 \( 1 + (-3.34 + 5.79i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.70 + 4.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.48 + 2.56i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.46T + 23T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.896 - 1.55i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.267 + 0.464i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.637 - 1.10i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.86 + 3.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.27 + 9.14i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.46 + 2.53i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.31 + 7.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.48 + 6.03i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.76 + 4.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.53T + 71T^{2} \)
73 \( 1 + (3.41 - 5.91i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.46 + 4.26i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.95 + 15.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.53 - 6.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.94 + 5.10i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.13450318487238054289103091145, −8.889269466959000913297106735206, −8.227908890722433285471341335044, −7.81625835152417030345213016629, −7.05693165514646336682925177679, −5.43377658960145931794812122090, −4.79290313203558727509707946338, −3.61257674737299318454270429313, −2.91563439141900880772643028047, −0.39721523854155235496622516594, 1.32435090818838941584201200827, 2.68413364127102804214475887955, 3.81612687123757283332972668483, 4.24761207862498467434086616485, 6.10046494323806040209121238405, 7.17259676002732270389840068012, 8.009730096945213514295080817666, 8.284370012124145211566281035275, 9.149823927474100646397234816429, 10.31978518780796663353751107030

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