Properties

Label 2-888-888.443-c1-0-33
Degree $2$
Conductor $888$
Sign $-0.0300 - 0.999i$
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  + (−1.27 − 0.620i)2-s + (1.50 + 0.859i)3-s + (1.23 + 1.57i)4-s + 2.29i·5-s + (−1.37 − 2.02i)6-s − 1.33i·7-s + (−0.587 − 2.76i)8-s + (1.52 + 2.58i)9-s + (1.42 − 2.91i)10-s + 2.88i·11-s + (0.497 + 3.42i)12-s − 1.66·13-s + (−0.828 + 1.69i)14-s + (−1.97 + 3.45i)15-s + (−0.968 + 3.88i)16-s + 0.260·17-s + ⋯
L(s)  = 1  + (−0.898 − 0.438i)2-s + (0.868 + 0.495i)3-s + (0.615 + 0.788i)4-s + 1.02i·5-s + (−0.562 − 0.826i)6-s − 0.505i·7-s + (−0.207 − 0.978i)8-s + (0.508 + 0.861i)9-s + (0.450 − 0.922i)10-s + 0.870i·11-s + (0.143 + 0.989i)12-s − 0.461·13-s + (−0.221 + 0.454i)14-s + (−0.509 + 0.891i)15-s + (−0.242 + 0.970i)16-s + 0.0630·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0300 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0300 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.0300 - 0.999i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ -0.0300 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.870355 + 0.896879i\)
\(L(\frac12)\) \(\approx\) \(0.870355 + 0.896879i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 + 0.620i)T \)
3 \( 1 + (-1.50 - 0.859i)T \)
37 \( 1 + (5.73 - 2.02i)T \)
good5 \( 1 - 2.29iT - 5T^{2} \)
7 \( 1 + 1.33iT - 7T^{2} \)
11 \( 1 - 2.88iT - 11T^{2} \)
13 \( 1 + 1.66T + 13T^{2} \)
17 \( 1 - 0.260T + 17T^{2} \)
19 \( 1 - 3.39iT - 19T^{2} \)
23 \( 1 + 0.960iT - 23T^{2} \)
29 \( 1 - 0.429iT - 29T^{2} \)
31 \( 1 - 2.48T + 31T^{2} \)
41 \( 1 + 5.80iT - 41T^{2} \)
43 \( 1 - 9.91iT - 43T^{2} \)
47 \( 1 + 5.41T + 47T^{2} \)
53 \( 1 + 7.55T + 53T^{2} \)
59 \( 1 - 4.57T + 59T^{2} \)
61 \( 1 - 1.49T + 61T^{2} \)
67 \( 1 + 8.82T + 67T^{2} \)
71 \( 1 + 7.07T + 71T^{2} \)
73 \( 1 + 5.34T + 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 + 5.04iT - 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 - 4.94iT - 97T^{2} \)
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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

−10.28674818465597998553965780601, −9.690030682953765958018679717386, −8.796498088479182120328102358411, −7.79459663427731566622968903193, −7.31330014379295345592796077199, −6.45497300137190500526993498670, −4.68745510454150183801039239973, −3.63794168604471085936590925358, −2.82056855571607092316693751406, −1.79397380636911954099454391243, 0.71975119278508386578999320705, 1.97486961015438687801719679185, 3.13193775121581739952775311041, 4.75639525120464302118801199101, 5.72679321286967005653422981797, 6.70905277001227809594013979506, 7.60352336371006160841937658200, 8.454615985656412597193465702907, 8.868248473007159252437147658546, 9.482345045177750263211216998588

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