Properties

Label 2-882-7.2-c3-0-2
Degree $2$
Conductor $882$
Sign $-0.991 - 0.126i$
Analytic cond. $52.0396$
Root an. cond. $7.21385$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (1.72 − 2.98i)5-s + 7.99·8-s + (3.44 + 5.96i)10-s + (18.0 + 31.2i)11-s − 10.2·13-s + (−8 + 13.8i)16-s + (−59.2 − 102. i)17-s + (−19.3 + 33.4i)19-s − 13.7·20-s − 72.2·22-s + (−18.1 + 31.3i)23-s + (56.5 + 97.9i)25-s + (10.2 − 17.7i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.154 − 0.266i)5-s + 0.353·8-s + (0.108 + 0.188i)10-s + (0.495 + 0.857i)11-s − 0.218·13-s + (−0.125 + 0.216i)16-s + (−0.845 − 1.46i)17-s + (−0.233 + 0.404i)19-s − 0.154·20-s − 0.700·22-s + (−0.164 + 0.284i)23-s + (0.452 + 0.783i)25-s + (0.0771 − 0.133i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(52.0396\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5557454245\)
\(L(\frac12)\) \(\approx\) \(0.5557454245\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-1.72 + 2.98i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-18.0 - 31.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 10.2T + 2.19e3T^{2} \)
17 \( 1 + (59.2 + 102. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (19.3 - 33.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (18.1 - 31.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 12.1T + 2.43e4T^{2} \)
31 \( 1 + (72.7 + 126. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-0.685 + 1.18i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 168T + 6.89e4T^{2} \)
43 \( 1 - 299.T + 7.95e4T^{2} \)
47 \( 1 + (251. - 435. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (312. + 541. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-21.1 - 36.5i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (219. - 380. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-381. - 661. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 1.02e3T + 3.57e5T^{2} \)
73 \( 1 + (-289. - 501. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (471. - 816. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 474.T + 5.71e5T^{2} \)
89 \( 1 + (410. - 711. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.10e3T + 9.12e5T^{2} \)
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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

−9.678083749285680145575845539030, −9.456499689879433138108150929811, −8.508307623726235293881156142822, −7.44840031337406686024750651657, −6.93371180572883851079238865532, −5.88367144442355866166647187246, −4.93459839390444368859145007906, −4.13110711761321918622436693131, −2.54606952673195964147098399612, −1.28160882334711784208958997830, 0.16799044867191723854931161457, 1.55599650052979551854300108149, 2.66977329168281945506787728865, 3.72986433601185443951104677809, 4.65284490165533835116164988548, 6.02357290737109661431326319523, 6.68879722135631091313959214588, 7.88162513661831890897960971361, 8.704867248165656432981303090362, 9.248307194798021306195398820695

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