Properties

Label 2-882-7.4-c3-0-1
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

\(L(2)\) \(\approx\) \(0.06543282456\)
\(L(\frac12)\) \(\approx\) \(0.06543282456\)
\(L(\frac{5}{2})\) 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

−9.867611361344047494224073112805, −9.214190609267769788922498777592, −8.416862759874529524168684061143, −7.65738530782225794183393466686, −6.65446219698949006125149972279, −5.42146974557432499100216895794, −4.50815662897138438606551624939, −3.50897636941868257749518980698, −2.40639646026928557675426577975, −1.05516544911179847314380282498, 0.02171972813948418476160604647, 1.69773132791504263229860553539, 2.98708632948703489660925369431, 4.32713596434689211206408774139, 5.10284611579858030000841392820, 6.30338605863651200564953700339, 7.18205303983618390362241017171, 7.51601502193497103526941039665, 8.703647013004659965841244356281, 9.505885511854461264337644169964

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