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Properties

Label 2-882-147.101-c1-0-5
Degree $2$
Conductor $882$
Sign $-0.263 - 0.964i$
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  + (0.930 + 0.365i)2-s + (0.733 + 0.680i)4-s + (2.63 + 1.79i)5-s + (−2.53 + 0.744i)7-s + (0.433 + 0.900i)8-s + (1.79 + 2.63i)10-s + (0.571 + 3.78i)11-s + (−5.36 + 4.28i)13-s + (−2.63 − 0.234i)14-s + (0.0747 + 0.997i)16-s + (−4.39 − 1.35i)17-s + (4.57 − 2.64i)19-s + (0.708 + 3.10i)20-s + (−0.852 + 3.73i)22-s + (−1.21 − 3.95i)23-s + ⋯
L(s)  = 1  + (0.658 + 0.258i)2-s + (0.366 + 0.340i)4-s + (1.17 + 0.801i)5-s + (−0.959 + 0.281i)7-s + (0.153 + 0.318i)8-s + (0.567 + 0.831i)10-s + (0.172 + 1.14i)11-s + (−1.48 + 1.18i)13-s + (−0.704 − 0.0627i)14-s + (0.0186 + 0.249i)16-s + (−1.06 − 0.329i)17-s + (1.04 − 0.606i)19-s + (0.158 + 0.693i)20-s + (−0.181 + 0.796i)22-s + (−0.254 − 0.824i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.263 - 0.964i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.263 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38260 + 1.81128i\)
\(L(\frac12)\) \(\approx\) \(1.38260 + 1.81128i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.930 - 0.365i)T \)
3 \( 1 \)
7 \( 1 + (2.53 - 0.744i)T \)
good5 \( 1 + (-2.63 - 1.79i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (-0.571 - 3.78i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (5.36 - 4.28i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (4.39 + 1.35i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (-4.57 + 2.64i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.21 + 3.95i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (-2.11 + 0.482i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-4.71 - 2.72i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.28 + 2.11i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (-2.26 + 1.09i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (-7.68 - 3.70i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-1.92 + 4.90i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (2.73 - 2.94i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (-5.23 + 3.57i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-4.87 - 5.25i)T + (-4.55 + 60.8i)T^{2} \)
67 \( 1 + (-1.99 + 3.45i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.06 - 2.06i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (10.9 - 4.30i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + (-3.29 - 5.70i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.93 + 4.93i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (17.0 + 2.56i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 - 10.9iT - 97T^{2} \)
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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

−10.11832253278575862893553119287, −9.685234456378723024605814408284, −8.988769871477427785474081545244, −7.27452558440345056272014398508, −6.84852692572870602355935240217, −6.23635873886548506565082246451, −5.08940155360936652012031999486, −4.26475783409716514286040906418, −2.62240533789771497593328275035, −2.32550682018691052256629596824, 0.856027626918180675909219835871, 2.41598393244737912243854891976, 3.34251668588760425788405905982, 4.61116962085239863682802153758, 5.66270390162978959811440000511, 5.97625884608444274194199713913, 7.16247079459719343629846210462, 8.256371064348254083523744050123, 9.430882542961694308741047484027, 9.810399021044448144273959219433

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