Properties

Label 2-888-296.179-c1-0-41
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: $\chi_{888} (475, \cdot )$
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

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

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