Properties

Label 2-882-7.5-c2-0-30
Degree $2$
Conductor $882$
Sign $-0.895 - 0.444i$
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  + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (−0.878 − 0.507i)5-s − 2.82·8-s + (−1.24 + 0.717i)10-s + (−5.12 − 8.87i)11-s + 8.95i·13-s + (−2.00 + 3.46i)16-s + (26.3 − 15.2i)17-s + (−13.9 − 8.06i)19-s + 2.02i·20-s − 14.4·22-s + (−3.36 + 5.82i)23-s + (−11.9 − 20.7i)25-s + (10.9 + 6.33i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.175 − 0.101i)5-s − 0.353·8-s + (−0.124 + 0.0717i)10-s + (−0.465 − 0.806i)11-s + 0.689i·13-s + (−0.125 + 0.216i)16-s + (1.54 − 0.894i)17-s + (−0.735 − 0.424i)19-s + 0.101i·20-s − 0.658·22-s + (−0.146 + 0.253i)23-s + (−0.479 − 0.830i)25-s + (0.421 + 0.243i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.895 - 0.444i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ -0.895 - 0.444i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6501330028\)
\(L(\frac12)\) \(\approx\) \(0.6501330028\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.878 + 0.507i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (5.12 + 8.87i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 8.95iT - 169T^{2} \)
17 \( 1 + (-26.3 + 15.2i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (13.9 + 8.06i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (3.36 - 5.82i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 30T + 841T^{2} \)
31 \( 1 + (43.4 - 25.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (15.4 - 26.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 7.10iT - 1.68e3T^{2} \)
43 \( 1 + 74.4T + 1.84e3T^{2} \)
47 \( 1 + (-50.4 - 29.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (35.4 + 61.4i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (0.426 - 0.246i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-2.48 - 1.43i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (13.5 + 23.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 50.6T + 5.04e3T^{2} \)
73 \( 1 + (-61.1 + 35.3i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (66.9 - 115. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 104. iT - 6.88e3T^{2} \)
89 \( 1 + (125. + 72.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 100. iT - 9.40e3T^{2} \)
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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

−9.584172903932958976588831388566, −8.735343288257157551669391273367, −7.84328383618496145771017985664, −6.82441761734371412336727588088, −5.71080775761467482705773313018, −4.99080483021600490046563195329, −3.82628814002723050118750941455, −2.98535274751071304275928654524, −1.67462447068852701387926046599, −0.18023651418260441292883668329, 1.84536494457328348153627609886, 3.34343254168462793801454324559, 4.13237682878854752850429058289, 5.42534375676695604602714255401, 5.85990298674309636420270965721, 7.19758335889708431287569027654, 7.70305435744782264298421568689, 8.494273308400616367846777135702, 9.616960020852716479475417842423, 10.32778408580568509649151889390

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