Properties

Label 2-888-296.43-c1-0-19
Degree $2$
Conductor $888$
Sign $0.883 - 0.469i$
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 − 0.615i)2-s − i·3-s + (1.24 + 1.56i)4-s + (−0.116 + 0.116i)5-s + (−0.615 + 1.27i)6-s + 4.08i·7-s + (−0.613 − 2.76i)8-s − 9-s + (0.220 − 0.0766i)10-s − 0.192i·11-s + (1.56 − 1.24i)12-s + (3.21 − 3.21i)13-s + (2.51 − 5.19i)14-s + (0.116 + 0.116i)15-s + (−0.919 + 3.89i)16-s + (−2.81 − 2.81i)17-s + ⋯
L(s)  = 1  + (−0.900 − 0.435i)2-s − 0.577i·3-s + (0.620 + 0.784i)4-s + (−0.0521 + 0.0521i)5-s + (−0.251 + 0.519i)6-s + 1.54i·7-s + (−0.217 − 0.976i)8-s − 0.333·9-s + (0.0697 − 0.0242i)10-s − 0.0579i·11-s + (0.452 − 0.358i)12-s + (0.892 − 0.892i)13-s + (0.672 − 1.38i)14-s + (0.0301 + 0.0301i)15-s + (−0.229 + 0.973i)16-s + (−0.681 − 0.681i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.883 - 0.469i$
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: \((2,\ 888,\ (\ :1/2),\ 0.883 - 0.469i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.869729 + 0.216646i\)
\(L(\frac12)\) \(\approx\) \(0.869729 + 0.216646i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 + 0.615i)T \)
3 \( 1 + iT \)
37 \( 1 + (2.55 - 5.52i)T \)
good5 \( 1 + (0.116 - 0.116i)T - 5iT^{2} \)
7 \( 1 - 4.08iT - 7T^{2} \)
11 \( 1 + 0.192iT - 11T^{2} \)
13 \( 1 + (-3.21 + 3.21i)T - 13iT^{2} \)
17 \( 1 + (2.81 + 2.81i)T + 17iT^{2} \)
19 \( 1 + (1.79 + 1.79i)T + 19iT^{2} \)
23 \( 1 + (2.66 - 2.66i)T - 23iT^{2} \)
29 \( 1 + (-7.17 - 7.17i)T + 29iT^{2} \)
31 \( 1 + (-6.96 - 6.96i)T + 31iT^{2} \)
41 \( 1 + 2.35iT - 41T^{2} \)
43 \( 1 + (-6.40 - 6.40i)T + 43iT^{2} \)
47 \( 1 + 2.36iT - 47T^{2} \)
53 \( 1 - 1.87iT - 53T^{2} \)
59 \( 1 + (-4.29 - 4.29i)T + 59iT^{2} \)
61 \( 1 + (7.77 + 7.77i)T + 61iT^{2} \)
67 \( 1 - 3.98iT - 67T^{2} \)
71 \( 1 - 12.3iT - 71T^{2} \)
73 \( 1 + 1.08iT - 73T^{2} \)
79 \( 1 + (-1.29 + 1.29i)T - 79iT^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 + (-5.63 + 5.63i)T - 89iT^{2} \)
97 \( 1 + (4.76 + 4.76i)T + 97iT^{2} \)
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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

−10.21145348848852952510747462313, −9.041387407447475863966166971272, −8.692182993693068094122540867296, −7.941849642391090724260912645217, −6.83881654675125182461951237081, −6.12444802610159861530111787254, −5.02485499895459719699197062766, −3.24222408539706369426126609275, −2.56324695308040664029632799110, −1.26813410625573710010870648154, 0.64211864034171302413663809684, 2.19995250446853810939939713257, 4.01609874553590997850818681759, 4.48229427763208951663207257080, 6.20587733941002073024435489608, 6.49376126466346220949972003115, 7.75391439178798667465531355757, 8.336124748573806856627658313024, 9.223146780174994509341231857153, 10.22449315343653672978812946116

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