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Properties

Label 2-882-21.17-c1-0-1
Degree $2$
Conductor $882$
Sign $-0.360 - 0.932i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (2.09 + 3.62i)5-s + 0.999i·8-s + (−3.62 − 2.09i)10-s + (2.59 + 1.5i)11-s + 2.44i·13-s + (−0.5 − 0.866i)16-s + (−0.507 + 0.878i)17-s + (0.878 − 0.507i)19-s + 4.18·20-s − 3·22-s + (3.67 − 2.12i)23-s + (−6.24 + 10.8i)25-s + (−1.22 − 2.12i)26-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.935 + 1.61i)5-s + 0.353i·8-s + (−1.14 − 0.661i)10-s + (0.783 + 0.452i)11-s + 0.679i·13-s + (−0.125 − 0.216i)16-s + (−0.123 + 0.213i)17-s + (0.201 − 0.116i)19-s + 0.935·20-s − 0.639·22-s + (0.766 − 0.442i)23-s + (−1.24 + 2.16i)25-s + (−0.240 − 0.416i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.360 - 0.932i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.360 - 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.773042 + 1.12698i\)
\(L(\frac12)\) \(\approx\) \(0.773042 + 1.12698i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-2.09 - 3.62i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.59 - 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + (0.507 - 0.878i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.878 + 0.507i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.67 + 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.24iT - 29T^{2} \)
31 \( 1 + (4.86 + 2.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.12 + 7.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.02T + 41T^{2} \)
43 \( 1 - 8.24T + 43T^{2} \)
47 \( 1 + (0.507 + 0.878i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.07 + 0.621i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.76 - 9.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.12 - 2.95i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.2iT - 71T^{2} \)
73 \( 1 + (7.24 + 4.18i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.62 - 9.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.16T + 83T^{2} \)
89 \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.76iT - 97T^{2} \)
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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

−10.43865342438985334178441942240, −9.368838446350377057389987073159, −9.107574074122320358785685494173, −7.56930589699605809743178640239, −6.95526050166521043133377650873, −6.36057124665616780447596345740, −5.49731475286494539753682897886, −3.99180190379906857045287788220, −2.69395525088350272172340010124, −1.72786654934789915777283165123, 0.849700686797922314823507543970, 1.78468534204426587935879380347, 3.27004519704467688194320881899, 4.58972807913839374776181632569, 5.46926404421387142128765173942, 6.32682612854922860878469981523, 7.58726291016116622761711588927, 8.536938497117716339856062161630, 9.095107231843855360916936928509, 9.640096602042570184053085569589

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