Properties

Label 2-882-7.2-c3-0-10
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 + (1.94 − 3.37i)5-s + 7.99·8-s + (3.89 + 6.75i)10-s + (−30.6 − 53.1i)11-s − 53.6·13-s + (−8 + 13.8i)16-s + (−16.0 − 27.7i)17-s + (−27.8 + 48.3i)19-s − 15.5·20-s + 122.·22-s + (−47.3 + 81.9i)23-s + (54.8 + 95.0i)25-s + (53.6 − 93.0i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.174 − 0.302i)5-s + 0.353·8-s + (0.123 + 0.213i)10-s + (−0.841 − 1.45i)11-s − 1.14·13-s + (−0.125 + 0.216i)16-s + (−0.228 − 0.396i)17-s + (−0.336 + 0.583i)19-s − 0.174·20-s + 1.18·22-s + (−0.428 + 0.742i)23-s + (0.439 + 0.760i)25-s + (0.405 − 0.701i)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: $\chi_{882} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ 0.0725 - 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9477732162\)
\(L(\frac12)\) \(\approx\) \(0.9477732162\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.94 + 3.37i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (30.6 + 53.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 53.6T + 2.19e3T^{2} \)
17 \( 1 + (16.0 + 27.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (27.8 - 48.3i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (47.3 - 81.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 138.T + 2.43e4T^{2} \)
31 \( 1 + (-66.3 - 114. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (74.6 - 129. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 427.T + 6.89e4T^{2} \)
43 \( 1 - 437.T + 7.95e4T^{2} \)
47 \( 1 + (28.5 - 49.3i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (131. + 228. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (225. + 391. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-289. + 501. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (154. + 268. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.05e3T + 3.57e5T^{2} \)
73 \( 1 + (-596. - 1.03e3i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (659. - 1.14e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.19e3T + 5.71e5T^{2} \)
89 \( 1 + (-116. + 201. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.60e3T + 9.12e5T^{2} \)
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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

−9.708105594945329745017685644741, −9.123598931589330606431319864171, −8.120389950164147583090808016070, −7.60364134333680865073358265895, −6.50994135632485623020398425925, −5.55600031177280125140090601833, −4.99971221706584919363118943696, −3.60468471299437802519719356598, −2.34432736295477954462089887187, −0.819270616589481615502111481695, 0.36357337352415531894280859600, 2.18269859558839755151234832493, 2.55018968910184049537498189922, 4.19243534058106496415949805305, 4.83971511523965663374739042027, 6.12489679631917629966206551459, 7.29883981885522250814564945099, 7.72734829137358865652338914359, 8.952494554967878068254215958780, 9.671186364710946874117563628692

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