Properties

Label 2-882-7.2-c3-0-28
Degree $2$
Conductor $882$
Sign $0.386 + 0.922i$
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 + (−7.61 + 13.1i)5-s + 7.99·8-s + (−15.2 − 26.3i)10-s + (−1 − 1.73i)11-s − 30.4·13-s + (−8 + 13.8i)16-s + (22.8 + 39.5i)17-s + (−76.1 + 131. i)19-s + 60.9·20-s + 3.99·22-s + (−15 + 25.9i)23-s + (−53.5 − 92.6i)25-s + (30.4 − 52.7i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.681 + 1.17i)5-s + 0.353·8-s + (−0.481 − 0.834i)10-s + (−0.0274 − 0.0474i)11-s − 0.649·13-s + (−0.125 + 0.216i)16-s + (0.325 + 0.564i)17-s + (−0.919 + 1.59i)19-s + 0.681·20-s + 0.0387·22-s + (−0.135 + 0.235i)23-s + (−0.427 − 0.741i)25-s + (0.229 − 0.397i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
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: \((2,\ 882,\ (\ :3/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.02609193429\)
\(L(\frac12)\) \(\approx\) \(0.02609193429\)
\(L(\frac{5}{2})\) 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 \( 1 + (7.61 - 13.1i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 30.4T + 2.19e3T^{2} \)
17 \( 1 + (-22.8 - 39.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (76.1 - 131. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (15 - 25.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 212T + 2.43e4T^{2} \)
31 \( 1 + (106. + 184. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (123 - 213. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 319.T + 6.89e4T^{2} \)
43 \( 1 + 284T + 7.95e4T^{2} \)
47 \( 1 + (30.4 - 52.7i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (274 + 474. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-335. - 580. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-258. + 448. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (326 + 564. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 770T + 3.57e5T^{2} \)
73 \( 1 + (-487. - 844. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (236 - 408. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 182.T + 5.71e5T^{2} \)
89 \( 1 + (357. - 619. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 304.T + 9.12e5T^{2} \)
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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

−9.830580219666995550956444993042, −8.303858417612162553731067765582, −8.045304897618168007436997613495, −6.95120432765467047228287483051, −6.43600263913128821405581532515, −5.36984866013650187197803752032, −4.12089353917022782885285906226, −3.22551462921797305953963307235, −1.83841889631958519726045116508, −0.009707296996795882412279657498, 0.886721651065065868683968136443, 2.26785669854818127806268129549, 3.46283750014384151986841379199, 4.69727183591611758149629490894, 5.00715450074054127180921348179, 6.68272054655359403922975010316, 7.54378143709658271497797886876, 8.595131882697620923693412661231, 8.863749605230486144460337694250, 9.892327199053087020565096864068

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