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Properties

Label 2-882-441.121-c1-0-7
Degree $2$
Conductor $882$
Sign $-0.283 - 0.958i$
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  + (0.900 − 0.433i)2-s + (−1.49 + 0.869i)3-s + (0.623 − 0.781i)4-s + (−1.18 + 0.366i)5-s + (−0.972 + 1.43i)6-s + (−1.17 − 2.37i)7-s + (0.222 − 0.974i)8-s + (1.48 − 2.60i)9-s + (−0.910 + 0.844i)10-s + (−0.189 + 2.53i)11-s + (−0.253 + 1.71i)12-s + (−0.325 + 4.34i)13-s + (−2.08 − 1.62i)14-s + (1.45 − 1.58i)15-s + (−0.222 − 0.974i)16-s + (−0.110 + 0.282i)17-s + ⋯
L(s)  = 1  + (0.637 − 0.306i)2-s + (−0.864 + 0.502i)3-s + (0.311 − 0.390i)4-s + (−0.530 + 0.163i)5-s + (−0.396 + 0.585i)6-s + (−0.444 − 0.896i)7-s + (0.0786 − 0.344i)8-s + (0.495 − 0.868i)9-s + (−0.287 + 0.267i)10-s + (−0.0572 + 0.763i)11-s + (−0.0732 + 0.494i)12-s + (−0.0902 + 1.20i)13-s + (−0.557 − 0.434i)14-s + (0.376 − 0.408i)15-s + (−0.0556 − 0.243i)16-s + (−0.0268 + 0.0685i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.283 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.283 - 0.958i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.283 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.477070 + 0.638708i\)
\(L(\frac12)\) \(\approx\) \(0.477070 + 0.638708i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.900 + 0.433i)T \)
3 \( 1 + (1.49 - 0.869i)T \)
7 \( 1 + (1.17 + 2.37i)T \)
good5 \( 1 + (1.18 - 0.366i)T + (4.13 - 2.81i)T^{2} \)
11 \( 1 + (0.189 - 2.53i)T + (-10.8 - 1.63i)T^{2} \)
13 \( 1 + (0.325 - 4.34i)T + (-12.8 - 1.93i)T^{2} \)
17 \( 1 + (0.110 - 0.282i)T + (-12.4 - 11.5i)T^{2} \)
19 \( 1 + (1.96 - 3.41i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.05 - 0.761i)T + (21.9 + 6.77i)T^{2} \)
29 \( 1 + (2.44 - 6.22i)T + (-21.2 - 19.7i)T^{2} \)
31 \( 1 + 7.00T + 31T^{2} \)
37 \( 1 + (-0.771 + 0.116i)T + (35.3 - 10.9i)T^{2} \)
41 \( 1 + (0.271 + 0.251i)T + (3.06 + 40.8i)T^{2} \)
43 \( 1 + (-0.766 + 0.711i)T + (3.21 - 42.8i)T^{2} \)
47 \( 1 + (9.39 - 4.52i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-8.54 - 1.28i)T + (50.6 + 15.6i)T^{2} \)
59 \( 1 + (0.142 + 0.623i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-6.32 - 7.92i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 + (-5.11 + 6.41i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-0.351 - 4.69i)T + (-72.1 + 10.8i)T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 + (-0.231 - 3.08i)T + (-82.0 + 12.3i)T^{2} \)
89 \( 1 + (-11.0 + 7.50i)T + (32.5 - 82.8i)T^{2} \)
97 \( 1 + (-3.37 - 5.84i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.57767462196659573126663631849, −9.809063688780247341232730529248, −9.034257768546945935704895125184, −7.32863847662624672184325332806, −6.97841405084973106504259178887, −5.95120759600054930978485889119, −4.86738806801223771420589227957, −4.10457772177218569485944944083, −3.45044983500950469591670759595, −1.58011809719177112172749096593, 0.34277358448901115779392036588, 2.37490258579603050604994804902, 3.50757918580337007630554870347, 4.82215976999426563407075634779, 5.58974371009276688928151135461, 6.21415032920349018171054527417, 7.17685547859676248246613648830, 8.027427764534928691145511678888, 8.802710072529709190682747325205, 10.05930466384894187460280054925

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