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Properties

Label 2-882-63.38-c1-0-36
Degree $2$
Conductor $882$
Sign $0.577 + 0.816i$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + (1.33 − 1.10i)3-s − 4-s + (1.94 − 3.36i)5-s + (1.10 + 1.33i)6-s − i·8-s + (0.577 − 2.94i)9-s + (3.36 + 1.94i)10-s + (−3.41 + 1.97i)11-s + (−1.33 + 1.10i)12-s + (2.46 − 1.42i)13-s + (−1.10 − 6.64i)15-s + 16-s + (−0.371 + 0.642i)17-s + (2.94 + 0.577i)18-s + (1.54 − 0.892i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.772 − 0.635i)3-s − 0.5·4-s + (0.870 − 1.50i)5-s + (0.449 + 0.546i)6-s − 0.353i·8-s + (0.192 − 0.981i)9-s + (1.06 + 0.615i)10-s + (−1.03 + 0.594i)11-s + (−0.386 + 0.317i)12-s + (0.684 − 0.395i)13-s + (−0.285 − 1.71i)15-s + 0.250·16-s + (−0.0899 + 0.155i)17-s + (0.693 + 0.136i)18-s + (0.354 − 0.204i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88470 - 0.975601i\)
\(L(\frac12)\) \(\approx\) \(1.88470 - 0.975601i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-1.33 + 1.10i)T \)
7 \( 1 \)
good5 \( 1 + (-1.94 + 3.36i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.41 - 1.97i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.46 + 1.42i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.371 - 0.642i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.54 + 0.892i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.41 - 3.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.50 + 1.44i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.51iT - 31T^{2} \)
37 \( 1 + (1.50 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.24 + 9.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.471 + 0.816i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.18T + 47T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.0211T + 59T^{2} \)
61 \( 1 - 2.46iT - 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 - 1.94iT - 71T^{2} \)
73 \( 1 + (-4.20 - 2.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 3.63T + 79T^{2} \)
83 \( 1 + (4.02 - 6.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.63 - 8.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-16.2 - 9.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.567155484379644325234715580045, −9.011767242869126846046517648930, −8.342258158888782573265160717691, −7.60351881780987224530947794474, −6.65497382340941102920509056603, −5.51139372885655646166685619555, −5.03762117114525627495438021215, −3.67362851023365637818965673565, −2.17523691815868291847584730992, −0.981096951439944518196630057216, 1.94628629956668636247032242094, 2.93739617380118159285072520449, 3.39331853410552546180700702295, 4.79870902171144893244323879375, 5.81711209606824803758356920134, 6.86484627977464331543801647211, 7.920756927483503347601123071469, 8.802885087598830432215309537623, 9.713133492530123730869281849949, 10.22818791195989970387779136816

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