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Properties

Label 2-882-441.130-c1-0-16
Degree $2$
Conductor $882$
Sign $0.999 - 0.0165i$
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  + (−0.826 + 0.563i)2-s + (−0.103 − 1.72i)3-s + (0.365 − 0.930i)4-s + (0.636 + 2.78i)5-s + (1.05 + 1.37i)6-s + (−2.56 + 0.630i)7-s + (0.222 + 0.974i)8-s + (−2.97 + 0.357i)9-s + (−2.09 − 1.94i)10-s + (−1.22 − 0.587i)11-s + (−1.64 − 0.535i)12-s + (5.87 − 4.00i)13-s + (1.76 − 1.96i)14-s + (4.75 − 1.38i)15-s + (−0.733 − 0.680i)16-s + (−0.581 − 1.48i)17-s + ⋯
L(s)  = 1  + (−0.584 + 0.398i)2-s + (−0.0597 − 0.998i)3-s + (0.182 − 0.465i)4-s + (0.284 + 1.24i)5-s + (0.432 + 0.559i)6-s + (−0.971 + 0.238i)7-s + (0.0786 + 0.344i)8-s + (−0.992 + 0.119i)9-s + (−0.663 − 0.615i)10-s + (−0.367 − 0.177i)11-s + (−0.475 − 0.154i)12-s + (1.63 − 1.11i)13-s + (0.472 − 0.526i)14-s + (1.22 − 0.358i)15-s + (−0.183 − 0.170i)16-s + (−0.141 − 0.359i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.999 - 0.0165i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.999 - 0.0165i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05995 + 0.00876658i\)
\(L(\frac12)\) \(\approx\) \(1.05995 + 0.00876658i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.826 - 0.563i)T \)
3 \( 1 + (0.103 + 1.72i)T \)
7 \( 1 + (2.56 - 0.630i)T \)
good5 \( 1 + (-0.636 - 2.78i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (1.22 + 0.587i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-5.87 + 4.00i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (0.581 + 1.48i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (3.13 + 5.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.44 - 5.57i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-6.47 - 0.976i)T + (27.7 + 8.54i)T^{2} \)
31 \( 1 + (-5.30 - 9.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.86 - 1.03i)T + (35.3 + 10.9i)T^{2} \)
41 \( 1 + (-4.85 - 1.49i)T + (33.8 + 23.0i)T^{2} \)
43 \( 1 + (2.91 - 0.899i)T + (35.5 - 24.2i)T^{2} \)
47 \( 1 + (-4.84 + 3.30i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-5.53 + 0.834i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (-1.22 + 0.379i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (1.50 + 3.84i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (4.40 + 7.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.22 - 1.53i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (0.196 - 2.62i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (-2.10 + 3.65i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.37 + 4.34i)T + (30.3 + 77.2i)T^{2} \)
89 \( 1 + (-9.11 - 6.21i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 + (1.85 + 3.21i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.29675564083013375924667666655, −9.074485932153397310349818569698, −8.435198886323287814755692968749, −7.44065570740807719506463279804, −6.57329012635527578280926439639, −6.33732729234973745438897723161, −5.32268460578164797394210455456, −3.14851863294727377765048544739, −2.70291917937256983247366095062, −0.908061949697540791397609400773, 0.922476250832120773264719639582, 2.56991626238265954293095041641, 4.01298229617489674013521373082, 4.37906667554808162480943557124, 5.88348126710769403171537494484, 6.44809513669686770625420759545, 8.109593352860175364541810878938, 8.763870583726078486405717236196, 9.247307374741071752639500614149, 10.10830774085013014334893041903

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