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Properties

Label 2-882-21.2-c2-0-8
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 + (−7.70 − 4.44i)5-s − 2.82i·8-s + (6.29 + 10.8i)10-s + (−15.4 + 8.89i)11-s − 2.58·13-s + (−2.00 + 3.46i)16-s + (−22.4 + 12.9i)17-s + (10 − 17.3i)19-s − 17.7i·20-s + 25.1·22-s + (15.4 + 8.89i)23-s + (27.0 + 46.9i)25-s + (3.16 + 1.82i)26-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.54 − 0.889i)5-s − 0.353i·8-s + (0.629 + 1.08i)10-s + (−1.40 + 0.808i)11-s − 0.198·13-s + (−0.125 + 0.216i)16-s + (−1.31 + 0.760i)17-s + (0.526 − 0.911i)19-s − 0.889i·20-s + 1.14·22-s + (0.670 + 0.386i)23-s + (1.08 + 1.87i)25-s + (0.121 + 0.0702i)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

\(L(\frac{3}{2})\) \(\approx\) \(0.5109499067\)
\(L(\frac12)\) \(\approx\) \(0.5109499067\)
\(L(2)\) 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 \( 1 + (7.70 + 4.44i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (15.4 - 8.89i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 2.58T + 169T^{2} \)
17 \( 1 + (22.4 - 12.9i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-10 + 17.3i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-15.4 - 8.89i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 11.9iT - 841T^{2} \)
31 \( 1 + (8.58 + 14.8i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (19 - 32.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 15.7iT - 1.68e3T^{2} \)
43 \( 1 + 43.4T + 1.84e3T^{2} \)
47 \( 1 + (-14.6 - 8.48i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-74.0 + 42.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-1.42 + 0.824i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-50.1 + 86.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (18.3 + 31.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 17.7iT - 5.04e3T^{2} \)
73 \( 1 + (-14.4 - 25.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (59.1 - 102. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 120. iT - 6.88e3T^{2} \)
89 \( 1 + (-120. - 69.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 44.4T + 9.40e3T^{2} \)
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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

−9.862525918226595988802379411041, −8.889394160990772573743138875410, −8.287653042285270424724613864477, −7.55368261450625210228439972575, −6.88603398632569772377034062243, −5.14885038616751665008928661444, −4.51722295279753782272751710224, −3.43942755917324408987394064785, −2.15126320118484902141268011810, −0.52565028014581074152692270118, 0.43116275614282985332850292183, 2.56144663408598434286614029512, 3.42549455030931859365103749252, 4.67444311490391642715677171306, 5.74298972203511128130222957624, 7.03688241102657216887211690564, 7.33136643574804240586446430115, 8.284891419791703685716417755578, 8.826819968831577698212855738467, 10.20656832571576266753126366737

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