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Properties

Label 2-882-147.104-c1-0-14
Degree $2$
Conductor $882$
Sign $-0.515 - 0.857i$
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  + (−0.781 − 0.623i)2-s + (0.222 + 0.974i)4-s + (−2.40 − 1.15i)5-s + (−0.0714 − 2.64i)7-s + (0.433 − 0.900i)8-s + (1.15 + 2.40i)10-s + (−1.90 − 1.52i)11-s + (1.46 + 1.17i)13-s + (−1.59 + 2.11i)14-s + (−0.900 + 0.433i)16-s + (−0.0538 + 0.236i)17-s − 0.476i·19-s + (0.593 − 2.59i)20-s + (0.542 + 2.37i)22-s + (−6.02 + 1.37i)23-s + ⋯
L(s)  = 1  + (−0.552 − 0.440i)2-s + (0.111 + 0.487i)4-s + (−1.07 − 0.517i)5-s + (−0.0269 − 0.999i)7-s + (0.153 − 0.318i)8-s + (0.365 + 0.759i)10-s + (−0.574 − 0.458i)11-s + (0.406 + 0.324i)13-s + (−0.425 + 0.564i)14-s + (−0.225 + 0.108i)16-s + (−0.0130 + 0.0572i)17-s − 0.109i·19-s + (0.132 − 0.581i)20-s + (0.115 + 0.506i)22-s + (−1.25 + 0.286i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.515 - 0.857i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.515 - 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0194366 + 0.0343615i\)
\(L(\frac12)\) \(\approx\) \(0.0194366 + 0.0343615i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.781 + 0.623i)T \)
3 \( 1 \)
7 \( 1 + (0.0714 + 2.64i)T \)
good5 \( 1 + (2.40 + 1.15i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (1.90 + 1.52i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.46 - 1.17i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.0538 - 0.236i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 0.476iT - 19T^{2} \)
23 \( 1 + (6.02 - 1.37i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-3.97 - 0.908i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 - 4.23iT - 31T^{2} \)
37 \( 1 + (1.24 - 5.46i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (2.77 + 1.33i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (0.992 - 0.477i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (4.28 - 5.37i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-1.18 + 0.271i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (8.79 - 4.23i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (8.96 + 2.04i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 0.386T + 67T^{2} \)
71 \( 1 + (-9.30 + 2.12i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (6.92 - 5.52i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 7.54T + 79T^{2} \)
83 \( 1 + (-0.608 - 0.763i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-4.28 - 5.37i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 16.9iT - 97T^{2} \)
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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

−9.678284554195992683545911838331, −8.534311173031100273869054996260, −8.085734855725183281855404855286, −7.31350590287312851695909653736, −6.29350488878041233136359585519, −4.80931890460246517858920074131, −4.00099201986534690636751994829, −3.11982801621614742824894704240, −1.36697148022014034336334660966, −0.02372372426936882544614749266, 2.12200864757820376184592541951, 3.31942852921972688130670136227, 4.54492545909633880784444380795, 5.67076579370748720656406437316, 6.47515742776181815739876117993, 7.54977570531416867586034296549, 8.061594597856336750724954869240, 8.828341091905400902868926436070, 9.848785964477708826384981943164, 10.57511735864223949157975851357

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