Properties

Label 2-882-63.25-c1-0-22
Degree $2$
Conductor $882$
Sign $-0.591 + 0.805i$
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  − 2-s + (0.5 − 1.65i)3-s + 4-s + (−2.18 + 3.78i)5-s + (−0.5 + 1.65i)6-s − 8-s + (−2.5 − 1.65i)9-s + (2.18 − 3.78i)10-s + (0.686 + 1.18i)11-s + (0.5 − 1.65i)12-s + (−1 − 1.73i)13-s + (5.18 + 5.51i)15-s + 16-s + (−0.686 + 1.18i)17-s + (2.5 + 1.65i)18-s + (−2.5 − 4.33i)19-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.288 − 0.957i)3-s + 0.5·4-s + (−0.977 + 1.69i)5-s + (−0.204 + 0.677i)6-s − 0.353·8-s + (−0.833 − 0.552i)9-s + (0.691 − 1.19i)10-s + (0.206 + 0.358i)11-s + (0.144 − 0.478i)12-s + (−0.277 − 0.480i)13-s + (1.33 + 1.42i)15-s + 0.250·16-s + (−0.166 + 0.288i)17-s + (0.589 + 0.390i)18-s + (−0.573 − 0.993i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.591 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.591 + 0.805i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.591 + 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.212352 - 0.419431i\)
\(L(\frac12)\) \(\approx\) \(0.212352 - 0.419431i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-0.5 + 1.65i)T \)
7 \( 1 \)
good5 \( 1 + (2.18 - 3.78i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.686 - 1.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.686 - 1.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.813 + 1.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.37 + 7.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.31 + 4.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.05 + 7.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-4.37 + 7.57i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 - 3.11T + 61T^{2} \)
67 \( 1 + 2.11T + 67T^{2} \)
71 \( 1 + 7.11T + 71T^{2} \)
73 \( 1 + (6.05 - 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 5.11T + 79T^{2} \)
83 \( 1 + (8.74 - 15.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.37 + 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.05 + 7.02i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.988522548724797040907127724231, −8.718501963835645491466014503024, −8.050000133186748098640191465932, −7.20522880343029496510321895500, −6.85177736274537856809789061654, −5.99163455981608958712422138955, −4.11883595147716406648718460210, −2.93794216901806269454716678468, −2.28577084803254638252556945450, −0.28275972629578904859996732317, 1.38059435506006490010756870256, 3.17659171318759441331865818555, 4.26668683659216218235463115051, 4.88131699890072881416083461128, 5.97922389491566618933190716438, 7.45212130261850003651368919876, 8.255647051388552304519508352129, 8.810065733564721948286372665220, 9.323766023920277870438810432577, 10.25654364590190934890442428196

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