Properties

Label 2-888-296.269-c1-0-55
Degree $2$
Conductor $888$
Sign $0.139 + 0.990i$
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  + (−1.40 − 0.147i)2-s + (0.866 − 0.5i)3-s + (1.95 + 0.413i)4-s + (2.32 − 1.34i)5-s + (−1.29 + 0.575i)6-s + (−0.950 − 1.64i)7-s + (−2.69 − 0.869i)8-s + (0.499 − 0.866i)9-s + (−3.46 + 1.54i)10-s + 2.68i·11-s + (1.90 − 0.619i)12-s + (−0.596 + 0.344i)13-s + (1.09 + 2.45i)14-s + (1.34 − 2.32i)15-s + (3.65 + 1.61i)16-s + (3.33 − 5.77i)17-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (0.499 − 0.288i)3-s + (0.978 + 0.206i)4-s + (1.03 − 0.599i)5-s + (−0.527 + 0.235i)6-s + (−0.359 − 0.622i)7-s + (−0.951 − 0.307i)8-s + (0.166 − 0.288i)9-s + (−1.09 + 0.488i)10-s + 0.810i·11-s + (0.548 − 0.178i)12-s + (−0.165 + 0.0955i)13-s + (0.292 + 0.656i)14-s + (0.346 − 0.599i)15-s + (0.914 + 0.404i)16-s + (0.808 − 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.139 + 0.990i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.139 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03180 - 0.897041i\)
\(L(\frac12)\) \(\approx\) \(1.03180 - 0.897041i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.147i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (1.31 - 5.93i)T \)
good5 \( 1 + (-2.32 + 1.34i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.950 + 1.64i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.68iT - 11T^{2} \)
13 \( 1 + (0.596 - 0.344i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.33 + 5.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.71 + 1.56i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.58T + 23T^{2} \)
29 \( 1 + 9.64iT - 29T^{2} \)
31 \( 1 - 9.33T + 31T^{2} \)
41 \( 1 + (1.95 + 3.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 8.09iT - 43T^{2} \)
47 \( 1 + 6.22T + 47T^{2} \)
53 \( 1 + (-1.10 - 0.635i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.29 - 5.36i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.74 + 1.58i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.41 - 1.97i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.20 + 12.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 + (1.61 + 2.80i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.47 - 4.31i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.292 + 0.506i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.33T + 97T^{2} \)
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

−9.879703272088484412368884714390, −9.373547117744651824615752790523, −8.269270857950454484028189448056, −7.54200697275212600454464397028, −6.75194001379345078883178049897, −5.82932440272878926742311997204, −4.56520164577187403793167473916, −3.04932188607589122783032572388, −2.06139047623065891295651798081, −0.881866919635188500824440219441, 1.60581379000218673151443455853, 2.68206551922755460182965289988, 3.52752623552124237676033147555, 5.55684734972252318907508655145, 6.01537473233991963300777124293, 6.94392041864376854864746365383, 8.117578386332606148175346694757, 8.634291741395819596683014017879, 9.580455981133323739202878850563, 10.19704355657545716163311968924

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