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Properties

Label 2-882-7.4-c1-0-3
Degree $2$
Conductor $882$
Sign $-0.968 - 0.250i$
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 + (−0.499 + 0.866i)4-s + (1 + 1.73i)5-s − 0.999·8-s + (−0.999 + 1.73i)10-s + (−2 + 3.46i)11-s − 6·13-s + (−0.5 − 0.866i)16-s + (−1 + 1.73i)17-s + (−2 − 3.46i)19-s − 1.99·20-s − 3.99·22-s + (4 + 6.92i)23-s + (0.500 − 0.866i)25-s + (−3 − 5.19i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.447 + 0.774i)5-s − 0.353·8-s + (−0.316 + 0.547i)10-s + (−0.603 + 1.04i)11-s − 1.66·13-s + (−0.125 − 0.216i)16-s + (−0.242 + 0.420i)17-s + (−0.458 − 0.794i)19-s − 0.447·20-s − 0.852·22-s + (0.834 + 1.44i)23-s + (0.100 − 0.173i)25-s + (−0.588 − 1.01i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.968 - 0.250i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.166053 + 1.30305i\)
\(L(\frac12)\) \(\approx\) \(0.166053 + 1.30305i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−10.22625386472067229882020250663, −9.859893420753569555582262348779, −8.801598430319248119453100341416, −7.63976104653277913101100762676, −7.09289247807004359454370261014, −6.37224329538730761845383534777, −5.14340923638633090293219468768, −4.60837865356506512641534448293, −3.08259396129458407694553315513, −2.19951601080366493466920784925, 0.52611535156862844656460187403, 2.13514168951973436735378160616, 3.08043975772074178389062491122, 4.51100270128946100487323118536, 5.13827866328819252008261178183, 5.99546887150647875495752095784, 7.16120562476150645619837818524, 8.304920417701320089062950205098, 9.005953684881011955540837994079, 9.863661167547984375619952395857

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