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Properties

Label 2-882-7.4-c1-0-11
Degree $2$
Conductor $882$
Sign $0.991 - 0.126i$
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 + (−0.5 − 0.866i)5-s − 0.999·8-s + (0.499 − 0.866i)10-s + (2.5 − 4.33i)11-s + (−0.5 − 0.866i)16-s + (2 − 3.46i)17-s + (4 + 6.92i)19-s + 0.999·20-s + 5·22-s + (−2 − 3.46i)23-s + (2 − 3.46i)25-s + 5·29-s + (1.5 − 2.59i)31-s + (0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s − 0.353·8-s + (0.158 − 0.273i)10-s + (0.753 − 1.30i)11-s + (−0.125 − 0.216i)16-s + (0.485 − 0.840i)17-s + (0.917 + 1.58i)19-s + 0.223·20-s + 1.06·22-s + (−0.417 − 0.722i)23-s + (0.400 − 0.692i)25-s + 0.928·29-s + (0.269 − 0.466i)31-s + (0.0883 − 0.153i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.991 - 0.126i$
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: \((2,\ 882,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82433 + 0.115772i\)
\(L(\frac12)\) \(\approx\) \(1.82433 + 0.115772i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4 - 6.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.5 + 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.5 + 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−10.01143729166675225316884430746, −9.164414113517100350406773077012, −8.255500785886550559870812573355, −7.77536487074137727204332385382, −6.49984525969903431241164747872, −5.91564569081852378850912014239, −4.89335366459859135369220866276, −3.90094578111485467999594722712, −2.95284197462116289950511356495, −0.947167516877525195914721387891, 1.33840474573420498753116503749, 2.64662095944537389539110956553, 3.71391614975541884787570827028, 4.61728997165275059376449943336, 5.57694513507731158627788298420, 6.80217048114000015377237886980, 7.35610229703135606974806560249, 8.624217823461401502501088680952, 9.519006735454344704222507572661, 10.11540704248499395603415540464

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