Properties

Label 2-888-296.43-c1-0-13
Degree $2$
Conductor $888$
Sign $0.977 - 0.211i$
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  + (−1.06 − 0.926i)2-s − i·3-s + (0.281 + 1.98i)4-s + (−2.48 + 2.48i)5-s + (−0.926 + 1.06i)6-s − 1.23i·7-s + (1.53 − 2.37i)8-s − 9-s + (4.96 − 0.351i)10-s − 5.59i·11-s + (1.98 − 0.281i)12-s + (−3.22 + 3.22i)13-s + (−1.14 + 1.32i)14-s + (2.48 + 2.48i)15-s + (−3.84 + 1.11i)16-s + (−2.22 − 2.22i)17-s + ⋯
L(s)  = 1  + (−0.755 − 0.655i)2-s − 0.577i·3-s + (0.140 + 0.990i)4-s + (−1.11 + 1.11i)5-s + (−0.378 + 0.436i)6-s − 0.468i·7-s + (0.542 − 0.840i)8-s − 0.333·9-s + (1.57 − 0.111i)10-s − 1.68i·11-s + (0.571 − 0.0812i)12-s + (−0.893 + 0.893i)13-s + (−0.306 + 0.353i)14-s + (0.642 + 0.642i)15-s + (−0.960 + 0.278i)16-s + (−0.540 − 0.540i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.977 - 0.211i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.977 - 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.640223 + 0.0684031i\)
\(L(\frac12)\) \(\approx\) \(0.640223 + 0.0684031i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.06 + 0.926i)T \)
3 \( 1 + iT \)
37 \( 1 + (-0.507 - 6.06i)T \)
good5 \( 1 + (2.48 - 2.48i)T - 5iT^{2} \)
7 \( 1 + 1.23iT - 7T^{2} \)
11 \( 1 + 5.59iT - 11T^{2} \)
13 \( 1 + (3.22 - 3.22i)T - 13iT^{2} \)
17 \( 1 + (2.22 + 2.22i)T + 17iT^{2} \)
19 \( 1 + (-5.67 - 5.67i)T + 19iT^{2} \)
23 \( 1 + (-2.40 + 2.40i)T - 23iT^{2} \)
29 \( 1 + (-4.45 - 4.45i)T + 29iT^{2} \)
31 \( 1 + (2.51 + 2.51i)T + 31iT^{2} \)
41 \( 1 - 10.0iT - 41T^{2} \)
43 \( 1 + (-7.24 - 7.24i)T + 43iT^{2} \)
47 \( 1 - 7.35iT - 47T^{2} \)
53 \( 1 + 0.173iT - 53T^{2} \)
59 \( 1 + (-3.79 - 3.79i)T + 59iT^{2} \)
61 \( 1 + (-1.24 - 1.24i)T + 61iT^{2} \)
67 \( 1 + 2.06iT - 67T^{2} \)
71 \( 1 + 12.8iT - 71T^{2} \)
73 \( 1 - 10.0iT - 73T^{2} \)
79 \( 1 + (0.543 - 0.543i)T - 79iT^{2} \)
83 \( 1 + 16.5T + 83T^{2} \)
89 \( 1 + (-9.27 + 9.27i)T - 89iT^{2} \)
97 \( 1 + (-5.10 - 5.10i)T + 97iT^{2} \)
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

−10.33160930686430409493964463449, −9.339734937017067612928864623271, −8.341442487182235373702630579924, −7.65440321603581864689677240698, −7.08295156591996145457715188811, −6.23954731069847663353052134179, −4.45159398566820597568476483935, −3.31711354127962642879056205978, −2.78887030541408885309173948537, −0.999726426541834875538242070666, 0.51269358777600134558033283158, 2.33456453883313709343771303771, 4.10414835010451748991595286289, 5.00969665602998989081761599228, 5.41690433291236490308654776297, 7.18361720235716209807836783472, 7.46772075543049092285658072884, 8.584576409948429846928719101581, 9.115499081311285711556461225325, 9.836345615312029659705187742785

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