Properties

Label 2-882-21.11-c2-0-14
Degree $2$
Conductor $882$
Sign $0.216 + 0.976i$
Analytic cond. $24.0327$
Root an. cond. $4.90232$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.707i)2-s + (0.999 − 1.73i)4-s + (−7.44 + 4.30i)5-s − 2.82i·8-s + (−6.08 + 10.5i)10-s + (2.44 + 1.41i)11-s − 12.1·13-s + (−2.00 − 3.46i)16-s + (22.3 + 12.9i)17-s + (−12.1 − 21.0i)19-s + 17.2i·20-s + 4·22-s + (36.7 − 21.2i)23-s + (24.5 − 42.4i)25-s + (−14.8 + 8.60i)26-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−1.48 + 0.860i)5-s − 0.353i·8-s + (−0.608 + 1.05i)10-s + (0.222 + 0.128i)11-s − 0.935·13-s + (−0.125 − 0.216i)16-s + (1.31 + 0.759i)17-s + (−0.640 − 1.10i)19-s + 0.860i·20-s + 0.181·22-s + (1.59 − 0.922i)23-s + (0.979 − 1.69i)25-s + (−0.573 + 0.330i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.216 + 0.976i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ 0.216 + 0.976i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.629790587\)
\(L(\frac12)\) \(\approx\) \(1.629790587\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (7.44 - 4.30i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.44 - 1.41i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 12.1T + 169T^{2} \)
17 \( 1 + (-22.3 - 12.9i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (12.1 + 21.0i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-36.7 + 21.2i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 15.5iT - 841T^{2} \)
31 \( 1 + (-12.1 + 21.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 25.8iT - 1.68e3T^{2} \)
43 \( 1 + 68T + 1.84e3T^{2} \)
47 \( 1 + (-59.5 + 34.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-35.5 - 20.5i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (59.5 + 34.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-48.6 - 84.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-52 + 90.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 70.7iT - 5.04e3T^{2} \)
73 \( 1 + (-30.4 + 52.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-10 - 17.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 + (81.9 - 47.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 158.T + 9.40e3T^{2} \)
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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

−10.10991717549725792545974062909, −8.879674922580019605503489388721, −7.87153849146626798985226161701, −7.13323047200826207068833947341, −6.44417612725968600279881627887, −5.06639345990871229796762689240, −4.22972552562895709574842534082, −3.34208739692758926976986093155, −2.47200357603658603120221075048, −0.52152547177415502502665984134, 1.09927522507250647586110911712, 3.06878141510135072579327248785, 3.85990459193185262313121855604, 4.87286000549799121534919358434, 5.41605214427582109396442017805, 6.88333625405914613280930340688, 7.55909838267731878384428800825, 8.220364674997089431425313785910, 9.064361564399302808766712891528, 10.10895304656171300984049316224

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