Properties

Label 2-888-296.221-c1-0-47
Degree $2$
Conductor $888$
Sign $0.882 + 0.471i$
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  + (1.39 + 0.248i)2-s − i·3-s + (1.87 + 0.692i)4-s − 1.62·5-s + (0.248 − 1.39i)6-s − 0.682·7-s + (2.43 + 1.43i)8-s − 9-s + (−2.25 − 0.403i)10-s − 1.65i·11-s + (0.692 − 1.87i)12-s + 6.88·13-s + (−0.950 − 0.169i)14-s + 1.62i·15-s + (3.04 + 2.59i)16-s − 5.88i·17-s + ⋯
L(s)  = 1  + (0.984 + 0.175i)2-s − 0.577i·3-s + (0.938 + 0.346i)4-s − 0.724·5-s + (0.101 − 0.568i)6-s − 0.258·7-s + (0.862 + 0.505i)8-s − 0.333·9-s + (−0.713 − 0.127i)10-s − 0.497i·11-s + (0.199 − 0.541i)12-s + 1.90·13-s + (−0.254 − 0.0453i)14-s + 0.418i·15-s + (0.760 + 0.649i)16-s − 1.42i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.882 + 0.471i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.882 + 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.68123 - 0.671058i\)
\(L(\frac12)\) \(\approx\) \(2.68123 - 0.671058i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.248i)T \)
3 \( 1 + iT \)
37 \( 1 + (0.243 + 6.07i)T \)
good5 \( 1 + 1.62T + 5T^{2} \)
7 \( 1 + 0.682T + 7T^{2} \)
11 \( 1 + 1.65iT - 11T^{2} \)
13 \( 1 - 6.88T + 13T^{2} \)
17 \( 1 + 5.88iT - 17T^{2} \)
19 \( 1 - 5.78T + 19T^{2} \)
23 \( 1 + 4.15iT - 23T^{2} \)
29 \( 1 - 5.64T + 29T^{2} \)
31 \( 1 - 9.48iT - 31T^{2} \)
41 \( 1 - 2.20T + 41T^{2} \)
43 \( 1 + 5.62T + 43T^{2} \)
47 \( 1 + 2.88T + 47T^{2} \)
53 \( 1 - 4.56iT - 53T^{2} \)
59 \( 1 + 6.60T + 59T^{2} \)
61 \( 1 - 0.499T + 61T^{2} \)
67 \( 1 - 10.9iT - 67T^{2} \)
71 \( 1 + 1.34T + 71T^{2} \)
73 \( 1 + 9.65T + 73T^{2} \)
79 \( 1 - 3.25iT - 79T^{2} \)
83 \( 1 + 12.4iT - 83T^{2} \)
89 \( 1 - 10.7iT - 89T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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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

−10.37712927377753357674313852153, −8.924098161875356783241544930275, −8.180859754274049319405491644073, −7.32398561615825572164814009245, −6.56901337395505469623907859354, −5.76202412465866746382093758727, −4.74422556173545455479530594438, −3.54527994059157361872004525188, −2.93621995220249929530393831784, −1.16370951564254095462972201006, 1.52048076066644153523543186454, 3.27549260706492082241336893845, 3.76662051508934514060813330922, 4.65220535706794411927295465429, 5.82256172418690866178723728679, 6.40397544140289224265030132152, 7.66635570289219312210202194109, 8.351359534778281285425861591022, 9.632639414468435013775218117631, 10.31748605815872615047611154548

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