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Properties

Label 2-882-21.2-c2-0-19
Degree $2$
Conductor $882$
Sign $0.935 + 0.354i$
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 + (5.25 + 3.03i)5-s − 2.82i·8-s + (−4.29 − 7.43i)10-s + (10.5 − 6.06i)11-s + 18.5·13-s + (−2.00 + 3.46i)16-s + (−9.44 + 5.45i)17-s + (10 − 17.3i)19-s + 12.1i·20-s − 17.1·22-s + (−10.5 − 6.06i)23-s + (5.91 + 10.2i)25-s + (−22.7 − 13.1i)26-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (1.05 + 0.606i)5-s − 0.353i·8-s + (−0.429 − 0.743i)10-s + (0.955 − 0.551i)11-s + 1.42·13-s + (−0.125 + 0.216i)16-s + (−0.555 + 0.320i)17-s + (0.526 − 0.911i)19-s + 0.606i·20-s − 0.780·22-s + (−0.457 − 0.263i)23-s + (0.236 + 0.409i)25-s + (−0.875 − 0.505i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.354i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.935 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.935 + 0.354i$
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: \((2,\ 882,\ (\ :1),\ 0.935 + 0.354i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.940848880\)
\(L(\frac12)\) \(\approx\) \(1.940848880\)
\(L(2)\) 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 \( 1 + (-5.25 - 3.03i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-10.5 + 6.06i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 18.5T + 169T^{2} \)
17 \( 1 + (9.44 - 5.45i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-10 + 17.3i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (10.5 + 6.06i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 41.8iT - 841T^{2} \)
31 \( 1 + (-12.5 - 21.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (19 - 32.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 60.6iT - 1.68e3T^{2} \)
43 \( 1 - 83.4T + 1.84e3T^{2} \)
47 \( 1 + (-14.6 - 8.48i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (81.4 - 47.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (50.4 - 29.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-7.83 + 13.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-66.3 - 114. i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 12.1iT - 5.04e3T^{2} \)
73 \( 1 + (38.4 + 66.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (16.8 - 29.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 60.5iT - 6.88e3T^{2} \)
89 \( 1 + (-4.13 - 2.38i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 188.T + 9.40e3T^{2} \)
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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

−9.905047450344223417945083238884, −9.040908723308059623118534434262, −8.560957533410972270921584746645, −7.32351143967010367854218037064, −6.28516530817176649382279244437, −5.98868135192814129898163880610, −4.30072062504350798494625686481, −3.22305854980543166554830758707, −2.14144054931658785290745086941, −0.982189537047271131950665002356, 1.14939723029524278417031192927, 1.91457550101544287865528097833, 3.62134305841585902634288915443, 4.84850129421103910859691393116, 5.91342172910428708435789203787, 6.38672369351720684700485393156, 7.50303093036779557467044216443, 8.485513603307890200183877413476, 9.279535550013544373602289654554, 9.616411288919367041946043958083

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