Properties

Label 2-882-7.6-c4-0-61
Degree $2$
Conductor $882$
Sign $-0.912 - 0.409i$
Analytic cond. $91.1723$
Root an. cond. $9.54841$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82·2-s + 8.00·4-s + 0.710i·5-s − 22.6·8-s − 2.01i·10-s − 151.·11-s − 260. i·13-s + 64.0·16-s − 385. i·17-s − 390. i·19-s + 5.68i·20-s + 428.·22-s + 177.·23-s + 624.·25-s + 737. i·26-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.500·4-s + 0.0284i·5-s − 0.353·8-s − 0.0201i·10-s − 1.25·11-s − 1.54i·13-s + 0.250·16-s − 1.33i·17-s − 1.08i·19-s + 0.0142i·20-s + 0.886·22-s + 0.335·23-s + 0.999·25-s + 1.09i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.912 - 0.409i$
Analytic conductor: \(91.1723\)
Root analytic conductor: \(9.54841\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :2),\ -0.912 - 0.409i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.2311998832\)
\(L(\frac12)\) \(\approx\) \(0.2311998832\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 0.710iT - 625T^{2} \)
11 \( 1 + 151.T + 1.46e4T^{2} \)
13 \( 1 + 260. iT - 2.85e4T^{2} \)
17 \( 1 + 385. iT - 8.35e4T^{2} \)
19 \( 1 + 390. iT - 1.30e5T^{2} \)
23 \( 1 - 177.T + 2.79e5T^{2} \)
29 \( 1 - 320.T + 7.07e5T^{2} \)
31 \( 1 - 1.34e3iT - 9.23e5T^{2} \)
37 \( 1 + 797.T + 1.87e6T^{2} \)
41 \( 1 - 815. iT - 2.82e6T^{2} \)
43 \( 1 + 2.16e3T + 3.41e6T^{2} \)
47 \( 1 + 4.28e3iT - 4.87e6T^{2} \)
53 \( 1 - 3.17e3T + 7.89e6T^{2} \)
59 \( 1 - 4.70e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.53e3iT - 1.38e7T^{2} \)
67 \( 1 + 4.09e3T + 2.01e7T^{2} \)
71 \( 1 - 2.25e3T + 2.54e7T^{2} \)
73 \( 1 - 4.65e3iT - 2.83e7T^{2} \)
79 \( 1 + 4.19e3T + 3.89e7T^{2} \)
83 \( 1 + 7.79e3iT - 4.74e7T^{2} \)
89 \( 1 - 9.46e3iT - 6.27e7T^{2} \)
97 \( 1 + 9.94e3iT - 8.85e7T^{2} \)
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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

−8.921667139731876803183748338969, −8.394598632800526980607825397573, −7.37561166091732352838327810715, −6.86201589544951280512520310694, −5.41453347474129537843072411074, −4.96324594642556271356790913802, −3.10647737106999495170724098661, −2.62697420434530297063870147519, −0.958869225167929989307498791344, −0.07571366206779459053786680440, 1.44522177720359180552738795983, 2.36011147685952423187654662312, 3.64917762434786145287767941705, 4.76041735526901602208643096786, 5.91462660514995331153210304986, 6.69238886922514729614678206804, 7.70188752908071278503395344238, 8.355298529187544875976439236523, 9.172836309541157981681816346735, 10.06761523971637973526847264086

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