Properties

Label 2-882-21.2-c2-0-22
Degree $2$
Conductor $882$
Sign $0.736 + 0.675i$
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 + (−3.67 − 2.12i)5-s + 2.82i·8-s + (−3 − 5.19i)10-s + (11.0 − 6.36i)11-s + 13-s + (−2.00 + 3.46i)16-s + (−14.6 + 8.48i)17-s + (11.5 − 19.9i)19-s − 8.48i·20-s + 18·22-s + (−14.6 − 8.48i)23-s + (−3.5 − 6.06i)25-s + (1.22 + 0.707i)26-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.734 − 0.424i)5-s + 0.353i·8-s + (−0.300 − 0.519i)10-s + (1.00 − 0.578i)11-s + 0.0769·13-s + (−0.125 + 0.216i)16-s + (−0.864 + 0.499i)17-s + (0.605 − 1.04i)19-s − 0.424i·20-s + 0.818·22-s + (−0.638 − 0.368i)23-s + (−0.140 − 0.242i)25-s + (0.0471 + 0.0271i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 + 0.675i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.736 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.736 + 0.675i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ 0.736 + 0.675i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.143163971\)
\(L(\frac12)\) \(\approx\) \(2.143163971\)
\(L(2)\) 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 \( 1 + (3.67 + 2.12i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-11.0 + 6.36i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - T + 169T^{2} \)
17 \( 1 + (14.6 - 8.48i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-11.5 + 19.9i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (14.6 + 8.48i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 33.9iT - 841T^{2} \)
31 \( 1 + (-23.5 - 40.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-27.5 + 47.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 46.6iT - 1.68e3T^{2} \)
43 \( 1 - 23T + 1.84e3T^{2} \)
47 \( 1 + (-3.67 - 2.12i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-44.0 + 25.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-73.4 + 42.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-52 + 90.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-48.5 - 84.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 97.5iT - 5.04e3T^{2} \)
73 \( 1 + (-32.5 - 56.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (56.5 - 97.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 29.6iT - 6.88e3T^{2} \)
89 \( 1 + (117. + 67.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 104T + 9.40e3T^{2} \)
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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

−9.736589071545016780405232082168, −8.704003528987477983749284884279, −8.245932835999674074381688178141, −7.11215120167335705718539066712, −6.41417459122697188194771653742, −5.41882174872234355810414217060, −4.28838311458721586253286546476, −3.80288116903528382436461741884, −2.39321785466991094162215090031, −0.62450697325769302954227057067, 1.30565479749577311215100566947, 2.67661858781656140749936850067, 3.80738305641596399747718637649, 4.38541112136983324777001557266, 5.62102557353525299232055117426, 6.60287619205933761807338443722, 7.33227200612471595361503334635, 8.267555547793185739722006058395, 9.447710485455163032201827727745, 10.04529156086119643048367014027

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