Properties

Label 2-888-296.221-c1-0-59
Degree $2$
Conductor $888$
Sign $0.508 + 0.860i$
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 + 0.200i)2-s − i·3-s + (1.91 − 0.561i)4-s + 2.44·5-s + (0.200 + 1.39i)6-s + 4.61·7-s + (−2.57 + 1.17i)8-s − 9-s + (−3.41 + 0.489i)10-s − 5.06i·11-s + (−0.561 − 1.91i)12-s + 0.712·13-s + (−6.45 + 0.924i)14-s − 2.44i·15-s + (3.37 − 2.15i)16-s − 7.27i·17-s + ⋯
L(s)  = 1  + (−0.989 + 0.141i)2-s − 0.577i·3-s + (0.959 − 0.280i)4-s + 1.09·5-s + (0.0818 + 0.571i)6-s + 1.74·7-s + (−0.910 + 0.413i)8-s − 0.333·9-s + (−1.08 + 0.154i)10-s − 1.52i·11-s + (−0.161 − 0.554i)12-s + 0.197·13-s + (−1.72 + 0.247i)14-s − 0.630i·15-s + (0.842 − 0.538i)16-s − 1.76i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.508 + 0.860i$
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: \((2,\ 888,\ (\ :1/2),\ 0.508 + 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27344 - 0.726629i\)
\(L(\frac12)\) \(\approx\) \(1.27344 - 0.726629i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.200i)T \)
3 \( 1 + iT \)
37 \( 1 + (6.04 - 0.650i)T \)
good5 \( 1 - 2.44T + 5T^{2} \)
7 \( 1 - 4.61T + 7T^{2} \)
11 \( 1 + 5.06iT - 11T^{2} \)
13 \( 1 - 0.712T + 13T^{2} \)
17 \( 1 + 7.27iT - 17T^{2} \)
19 \( 1 - 0.0331T + 19T^{2} \)
23 \( 1 - 4.39iT - 23T^{2} \)
29 \( 1 - 0.282T + 29T^{2} \)
31 \( 1 - 7.20iT - 31T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + 5.99T + 43T^{2} \)
47 \( 1 - 8.51T + 47T^{2} \)
53 \( 1 - 3.34iT - 53T^{2} \)
59 \( 1 - 2.99T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 - 12.4iT - 67T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 - 5.32T + 73T^{2} \)
79 \( 1 + 1.15iT - 79T^{2} \)
83 \( 1 - 3.97iT - 83T^{2} \)
89 \( 1 + 4.16iT - 89T^{2} \)
97 \( 1 + 13.9iT - 97T^{2} \)
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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

−9.963231365711371153452129527643, −8.795260432039089956554827179235, −8.557014629864520416457703085112, −7.52201603186722001667868868799, −6.78201340750826123504658030980, −5.59184642873075236672218908704, −5.23762908624972893101911480926, −3.07556585139396660750096973012, −1.91772663772722623226071198137, −1.05497614245753450383576681499, 1.72754889946985183053644519865, 2.11603816544785518342684823663, 3.97920953606266754973573059261, 5.01627849014649737398853563718, 5.96904026756672838424228758227, 6.99425342221498760732235621138, 8.084534946156492569441544827286, 8.572947746992434257970005681250, 9.549459481666894876872891977138, 10.32283653306395903475890556509

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