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Properties

Label 2-882-7.4-c3-0-38
Degree $2$
Conductor $882$
Sign $0.0725 + 0.997i$
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 + (3.53 + 6.12i)5-s − 7.99·8-s + (−7.07 + 12.2i)10-s + (−20 + 34.6i)11-s − 63.6·13-s + (−8 − 13.8i)16-s + (−0.707 + 1.22i)17-s + (−5.65 − 9.79i)19-s − 28.2·20-s − 80·22-s + (−34 − 58.8i)23-s + (37.5 − 64.9i)25-s + (−63.6 − 110. i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.316 + 0.547i)5-s − 0.353·8-s + (−0.223 + 0.387i)10-s + (−0.548 + 0.949i)11-s − 1.35·13-s + (−0.125 − 0.216i)16-s + (−0.0100 + 0.0174i)17-s + (−0.0683 − 0.118i)19-s − 0.316·20-s − 0.775·22-s + (−0.308 − 0.533i)23-s + (0.299 − 0.519i)25-s + (−0.480 − 0.831i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.0725 + 0.997i$
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: \((2,\ 882,\ (\ :3/2),\ 0.0725 + 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2304491838\)
\(L(\frac12)\) \(\approx\) \(0.2304491838\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.53 - 6.12i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (20 - 34.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 63.6T + 2.19e3T^{2} \)
17 \( 1 + (0.707 - 1.22i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (5.65 + 9.79i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (34 + 58.8i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 110T + 2.43e4T^{2} \)
31 \( 1 + (-59.3 + 102. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-10 - 17.3i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 49.4T + 6.89e4T^{2} \)
43 \( 1 + 340T + 7.95e4T^{2} \)
47 \( 1 + (45.2 + 78.3i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (314 - 543. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-438. + 759. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (458. + 794. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (270 - 467. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 420T + 3.57e5T^{2} \)
73 \( 1 + (144. - 251. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-380 - 658. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 944.T + 5.71e5T^{2} \)
89 \( 1 + (576. + 998. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 502.T + 9.12e5T^{2} \)
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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

−9.791610276452341112978552464677, −8.512962360752211471047697243036, −7.66695114140624307260522699153, −6.93050031375399063111808497326, −6.21268011240786917290888159233, −5.02671289481431851270752577012, −4.46712563938481778299090351592, −2.98880761201439101731499402306, −2.13880586113806503739074490906, −0.05156449302397461625119695444, 1.25438283622205375652381472402, 2.50076238384072607333124291481, 3.43573206677120531196063393186, 4.76263510849454816735267671287, 5.29528577687905139123514401920, 6.29381030339175686200967985491, 7.46313369974722590199799050629, 8.433944051900238718188179952479, 9.232367808974950568269990335208, 10.06982561943970258776241911061

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