Properties

Label 2-882-7.4-c3-0-42
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: $\chi_{882} (361, \cdot )$
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

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

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