Properties

Label 2-888-296.179-c1-0-30
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

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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: $\chi_{888} (475, \cdot )$
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 \( 1 + (-5.10 + 5.10i)T - 97iT^{2} \)
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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

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

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