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Properties

Label 2-882-147.104-c1-0-1
Degree $2$
Conductor $882$
Sign $0.485 - 0.874i$
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 + (1.04 + 0.505i)5-s + (−2.62 + 0.311i)7-s + (0.433 − 0.900i)8-s + (−0.505 − 1.04i)10-s + (−2.62 − 2.09i)11-s + (2.75 + 2.19i)13-s + (2.24 + 1.39i)14-s + (−0.900 + 0.433i)16-s + (−0.165 + 0.722i)17-s + 6.51i·19-s + (−0.259 + 1.13i)20-s + (0.747 + 3.27i)22-s + (8.22 − 1.87i)23-s + ⋯
L(s)  = 1  + (−0.552 − 0.440i)2-s + (0.111 + 0.487i)4-s + (0.469 + 0.225i)5-s + (−0.993 + 0.117i)7-s + (0.153 − 0.318i)8-s + (−0.159 − 0.331i)10-s + (−0.791 − 0.631i)11-s + (0.763 + 0.608i)13-s + (0.600 + 0.372i)14-s + (−0.225 + 0.108i)16-s + (−0.0400 + 0.175i)17-s + 1.49i·19-s + (−0.0579 + 0.253i)20-s + (0.159 + 0.698i)22-s + (1.71 − 0.391i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.485 - 0.874i$
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: \((2,\ 882,\ (\ :1/2),\ 0.485 - 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.771727 + 0.453937i\)
\(L(\frac12)\) \(\approx\) \(0.771727 + 0.453937i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 - 0.311i)T \)
good5 \( 1 + (-1.04 - 0.505i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (2.62 + 2.09i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-2.75 - 2.19i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.165 - 0.722i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 6.51iT - 19T^{2} \)
23 \( 1 + (-8.22 + 1.87i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-2.13 - 0.488i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 - 8.50iT - 31T^{2} \)
37 \( 1 + (2.22 - 9.73i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-0.840 - 0.404i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (10.1 - 4.88i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (6.85 - 8.59i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-5.82 + 1.33i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (12.5 - 6.05i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-4.31 - 0.985i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 7.58T + 67T^{2} \)
71 \( 1 + (2.95 - 0.673i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-11.1 + 8.90i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + (1.64 + 2.06i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-3.51 - 4.40i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 3.07iT - 97T^{2} \)
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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

−10.31912091631206552601595084975, −9.546730314628724543176675434801, −8.669502225708264887076460438501, −8.025844765262297146492448660179, −6.67183706946100470209182677686, −6.24879517168725135533733969476, −4.99426957321165718733350575069, −3.51134000600169514277193544809, −2.85262345840950301622648031440, −1.39204239497114314021913586098, 0.53901534322452255717136156237, 2.26737059280218523200002538860, 3.45495058953397011702816274102, 4.98224137206486097635068920235, 5.66180274102583136163427647570, 6.75049032615306732338116323060, 7.31001375830341324585588633791, 8.398402897099323833073620026287, 9.291604950565445138487794948914, 9.733137229427726274495300767084

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