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Properties

Label 2-882-7.4-c3-0-14
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 + (3 + 5.19i)5-s + 7.99·8-s + (6 − 10.3i)10-s + (6 − 10.3i)11-s + 38·13-s + (−8 − 13.8i)16-s + (−63 + 109. i)17-s + (−10 − 17.3i)19-s − 24·20-s − 24·22-s + (84 + 145. i)23-s + (44.5 − 77.0i)25-s + (−38 − 65.8i)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 + (0.164 − 0.284i)11-s + 0.810·13-s + (−0.125 − 0.216i)16-s + (−0.898 + 1.55i)17-s + (−0.120 − 0.209i)19-s − 0.268·20-s − 0.232·22-s + (0.761 + 1.31i)23-s + (0.355 − 0.616i)25-s + (−0.286 − 0.496i)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: $\chi_{882} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.302297057\)
\(L(\frac12)\) \(\approx\) \(1.302297057\)
\(L(\frac{5}{2})\) 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 \( 1 + (-6 + 10.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 38T + 2.19e3T^{2} \)
17 \( 1 + (63 - 109. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (10 + 17.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-84 - 145. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 30T + 2.43e4T^{2} \)
31 \( 1 + (-44 + 76.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (127 + 219. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 42T + 6.89e4T^{2} \)
43 \( 1 + 52T + 7.95e4T^{2} \)
47 \( 1 + (48 + 83.1i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-99 + 171. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (330 - 571. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-269 - 465. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (442 - 765. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 792T + 3.57e5T^{2} \)
73 \( 1 + (109 - 188. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-260 - 450. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 492T + 5.71e5T^{2} \)
89 \( 1 + (-405 - 701. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.15e3T + 9.12e5T^{2} \)
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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

−10.08849055691783024304383864067, −8.970097497878689072455962374742, −8.550519282306824919977922489913, −7.42372414734217084453924686056, −6.47530143900682434657446490611, −5.64110396456652146859397301841, −4.24443504361926402267972515234, −3.43636011241455733264699520589, −2.27096743061844040786183254662, −1.19309288129914883362263952543, 0.41471334992363735395026468683, 1.63137306130518107906551157691, 3.07347449664955692159811700554, 4.56806458865887060313623383011, 5.12084185139005356338649707704, 6.35385717241385860772945827279, 6.91083533224376712675755558485, 7.957296687777046574672673836626, 8.928160565967096185830645516997, 9.219011140302947642214890488110

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