↑

Properties

Label 2-888-8.5-c1-0-42
Degree $2$
Conductor $888$
Sign $0.790 - 0.612i$
Analytic cond. $7.09071$
Root an. cond. $2.66283$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.38 − 0.308i)2-s + i·3-s + (1.81 − 0.850i)4-s + 3.14i·5-s + (0.308 + 1.38i)6-s + 3.55·7-s + (2.23 − 1.73i)8-s − 9-s + (0.968 + 4.33i)10-s − 1.80i·11-s + (0.850 + 1.81i)12-s − 4.40i·13-s + (4.90 − 1.09i)14-s − 3.14·15-s + (2.55 − 3.07i)16-s + 2.28·17-s + ⋯
L(s)  = 1  + (0.975 − 0.217i)2-s + 0.577i·3-s + (0.905 − 0.425i)4-s + 1.40i·5-s + (0.125 + 0.563i)6-s + 1.34·7-s + (0.790 − 0.612i)8-s − 0.333·9-s + (0.306 + 1.37i)10-s − 0.545i·11-s + (0.245 + 0.522i)12-s − 1.22i·13-s + (1.31 − 0.292i)14-s − 0.811·15-s + (0.638 − 0.769i)16-s + 0.554·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.790 - 0.612i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.790 - 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.12194 + 1.06763i\)
\(L(\frac12)\) \(\approx\) \(3.12194 + 1.06763i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.308i)T \)
3 \( 1 - iT \)
37 \( 1 - iT \)
good5 \( 1 - 3.14iT - 5T^{2} \)
7 \( 1 - 3.55T + 7T^{2} \)
11 \( 1 + 1.80iT - 11T^{2} \)
13 \( 1 + 4.40iT - 13T^{2} \)
17 \( 1 - 2.28T + 17T^{2} \)
19 \( 1 - 6.35iT - 19T^{2} \)
23 \( 1 + 8.28T + 23T^{2} \)
29 \( 1 - 2.44iT - 29T^{2} \)
31 \( 1 - 4.63T + 31T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 2.53iT - 43T^{2} \)
47 \( 1 + 3.74T + 47T^{2} \)
53 \( 1 - 8.33iT - 53T^{2} \)
59 \( 1 + 13.2iT - 59T^{2} \)
61 \( 1 + 10.1iT - 61T^{2} \)
67 \( 1 + 3.73iT - 67T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 - 0.284T + 79T^{2} \)
83 \( 1 + 6.02iT - 83T^{2} \)
89 \( 1 - 16.5T + 89T^{2} \)
97 \( 1 + 7.81T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.36322019315924876861212104406, −9.997944834705310503224858841011, −8.056951337170991942012903683956, −7.86184908796279399902491616705, −6.45220042227269792484768082259, −5.75087529116466525995700404179, −4.90245130930751091532651083912, −3.69942155136822740008239290359, −3.08772073715530338732605358238, −1.80476601594685629831129637363, 1.42895275543527846291252720073, 2.24952635667151506939127199252, 4.12234801802239905342954287117, 4.73571781048082571625325885615, 5.40388524897951886743497285284, 6.53191986444882029020695790744, 7.46521095316724507494552515087, 8.234148654943086077120193314921, 8.849364296141109997431353747727, 10.10012423965924597320356800326

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