Properties

Label 2-888-888.221-c0-0-8
Degree $2$
Conductor $888$
Sign $-0.382 + 0.923i$
Analytic cond. $0.443169$
Root an. cond. $0.665709$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.923 + 0.382i)2-s + i·3-s + (0.707 − 0.707i)4-s − 1.84i·5-s + (−0.382 − 0.923i)6-s − 1.41·7-s + (−0.382 + 0.923i)8-s − 9-s + (0.707 + 1.70i)10-s + (0.707 + 0.707i)12-s + (1.30 − 0.541i)14-s + 1.84·15-s − i·16-s − 1.84·17-s + (0.923 − 0.382i)18-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)2-s + i·3-s + (0.707 − 0.707i)4-s − 1.84i·5-s + (−0.382 − 0.923i)6-s − 1.41·7-s + (−0.382 + 0.923i)8-s − 9-s + (0.707 + 1.70i)10-s + (0.707 + 0.707i)12-s + (1.30 − 0.541i)14-s + 1.84·15-s − i·16-s − 1.84·17-s + (0.923 − 0.382i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(0.443169\)
Root analytic conductor: \(0.665709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :0),\ -0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2183796644\)
\(L(\frac12)\) \(\approx\) \(0.2183796644\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 - 0.382i)T \)
3 \( 1 - iT \)
37 \( 1 + iT \)
good5 \( 1 + 1.84iT - T^{2} \)
7 \( 1 + 1.41T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + 1.84T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 0.765T + T^{2} \)
29 \( 1 + 0.765iT - T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 0.765iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.41iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.41T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 0.765T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.689384784473319530787777328119, −9.222390307219324859557599049393, −8.753052392805460501136440568403, −7.890908817361212131887256397748, −6.48874940003770304535413464326, −5.78988419015930897184195670462, −4.81536136812625801071439576600, −3.89872408514732513703421642574, −2.27699718929952626405957323285, −0.26108003368098071012425748435, 2.15061405614324405878411597818, 2.85803003339248788004520323942, 3.67491579379230410521006489274, 6.11879991170132349231115745870, 6.67052847548823380924408672396, 6.99155996966964846967913420319, 7.971564249547002270391750911766, 8.982784984913153037418617894730, 9.853637499453355229819892607982, 10.60541286535701132142188303327

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