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Properties

Label 2-882-7.4-c3-0-20
Degree $2$
Conductor $882$
Sign $0.968 + 0.250i$
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 + 5.19i)5-s + 7.99·8-s + (6 − 10.3i)10-s + (−15 + 25.9i)11-s − 2·13-s + (−8 − 13.8i)16-s + (33 − 57.1i)17-s + (−26 − 45.0i)19-s − 24·20-s + 60·22-s + (−57 − 98.7i)23-s + (44.5 − 77.0i)25-s + (2 + 3.46i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.268 + 0.464i)5-s + 0.353·8-s + (0.189 − 0.328i)10-s + (−0.411 + 0.712i)11-s − 0.0426·13-s + (−0.125 − 0.216i)16-s + (0.470 − 0.815i)17-s + (−0.313 − 0.543i)19-s − 0.268·20-s + 0.581·22-s + (−0.516 − 0.895i)23-s + (0.355 − 0.616i)25-s + (0.0150 + 0.0261i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.968 + 0.250i$
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: \((2,\ 882,\ (\ :3/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.571451570\)
\(L(\frac12)\) \(\approx\) \(1.571451570\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3 - 5.19i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (15 - 25.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 2T + 2.19e3T^{2} \)
17 \( 1 + (-33 + 57.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (26 + 45.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (57 + 98.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 72T + 2.43e4T^{2} \)
31 \( 1 + (98 - 169. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-143 - 247. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 378T + 6.89e4T^{2} \)
43 \( 1 - 164T + 7.95e4T^{2} \)
47 \( 1 + (114 + 197. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-174 + 301. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (174 - 301. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (53 + 91.7i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (298 - 516. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 630T + 3.57e5T^{2} \)
73 \( 1 + (521 - 902. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-44 - 76.2i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.44e3T + 5.71e5T^{2} \)
89 \( 1 + (-687 - 1.18e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 34T + 9.12e5T^{2} \)
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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

−9.911747274767855155764208235351, −9.020389069453734014911148294196, −8.127124506809661961081667218024, −7.21936722070184793903233450246, −6.42536721549335911626879342454, −5.13686654841852371471753554061, −4.27023901324165010245570402258, −2.92515121386079999367154381441, −2.24259836233202707629147241309, −0.76325493790958924150143379966, 0.69072460595655420548338110325, 1.92950797605858937262719438194, 3.47223936446646124699993772583, 4.58023555346908045159527297408, 5.80989874921359243988112610296, 6.00061347457366707799859806153, 7.51447013246395776466650983737, 7.960002754200240815875550248986, 8.977295028690960647299524556894, 9.538538967930463187312886892441

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