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Properties

Label 2-882-9.7-c1-0-5
Degree $2$
Conductor $882$
Sign $-0.493 - 0.869i$
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 + (−0.306 + 1.70i)3-s + (−0.499 − 0.866i)4-s + (1.62 + 2.82i)5-s + (1.32 + 1.11i)6-s − 0.999·8-s + (−2.81 − 1.04i)9-s + 3.25·10-s + (−2.81 + 4.87i)11-s + (1.62 − 0.586i)12-s + (0.613 + 1.06i)13-s + (−5.31 + 1.91i)15-s + (−0.5 + 0.866i)16-s − 5.90·17-s + (−2.31 + 1.91i)18-s + 2.64·19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.177 + 0.984i)3-s + (−0.249 − 0.433i)4-s + (0.728 + 1.26i)5-s + (0.540 + 0.456i)6-s − 0.353·8-s + (−0.937 − 0.348i)9-s + 1.03·10-s + (−0.847 + 1.46i)11-s + (0.470 − 0.169i)12-s + (0.170 + 0.294i)13-s + (−1.37 + 0.493i)15-s + (−0.125 + 0.216i)16-s − 1.43·17-s + (−0.544 + 0.450i)18-s + 0.606·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.493 - 0.869i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.493 - 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.674025 + 1.15795i\)
\(L(\frac12)\) \(\approx\) \(0.674025 + 1.15795i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.306 - 1.70i)T \)
7 \( 1 \)
good5 \( 1 + (-1.62 - 2.82i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.81 - 4.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.613 - 1.06i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.90T + 17T^{2} \)
19 \( 1 - 2.64T + 19T^{2} \)
23 \( 1 + (3.31 + 5.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2 + 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.613 + 1.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (-2.95 - 5.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.81 - 6.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.29 - 9.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (-7.43 - 12.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.24 + 3.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.81 - 4.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 - 11.1T + 73T^{2} \)
79 \( 1 + (1.68 - 2.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.87 + 6.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 8.97T + 89T^{2} \)
97 \( 1 + (1.53 - 2.65i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.34544487241907674473787004066, −9.900147035040756845405237831382, −9.234660699390558046749927934305, −7.925418268679513075732256968469, −6.64766488407922834603128431178, −6.06926534958121904342895587013, −4.81896044673631661531240435730, −4.23835013706424996954253663111, −2.81651781903302358397688872471, −2.27054593071924833723203692791, 0.55114083504725354834598234973, 2.01642643235596971097763209076, 3.40042407276845395119462250296, 4.99483950126122777966754099692, 5.54133015529419936597377388503, 6.20123092367557000256568618538, 7.24756296354324378857785704926, 8.336661030606510141526473152271, 8.567952681265119763899935853572, 9.581878558762021894580887981371

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