Properties

Label 2-888-296.179-c1-0-14
Degree $2$
Conductor $888$
Sign $0.320 - 0.947i$
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 + (0.125 + 0.125i)5-s + (0.615 − 1.27i)6-s − 2.31i·7-s + (−0.619 − 2.75i)8-s − 9-s + (−0.0827 − 0.237i)10-s + 2.93i·11-s + (−1.56 + 1.24i)12-s + (1.37 + 1.37i)13-s + (−1.42 + 2.95i)14-s + (−0.125 + 0.125i)15-s + (−0.909 + 3.89i)16-s + (−5.24 + 5.24i)17-s + ⋯
L(s)  = 1  + (−0.900 − 0.434i)2-s + 0.577i·3-s + (0.621 + 0.783i)4-s + (0.0561 + 0.0561i)5-s + (0.251 − 0.519i)6-s − 0.875i·7-s + (−0.218 − 0.975i)8-s − 0.333·9-s + (−0.0261 − 0.0750i)10-s + 0.883i·11-s + (−0.452 + 0.358i)12-s + (0.380 + 0.380i)13-s + (−0.380 + 0.788i)14-s + (−0.0324 + 0.0324i)15-s + (−0.227 + 0.973i)16-s + (−1.27 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.320 - 0.947i$
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: \((2,\ 888,\ (\ :1/2),\ 0.320 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.703060 + 0.504230i\)
\(L(\frac12)\) \(\approx\) \(0.703060 + 0.504230i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.00 + 0.943i)T \)
good5 \( 1 + (-0.125 - 0.125i)T + 5iT^{2} \)
7 \( 1 + 2.31iT - 7T^{2} \)
11 \( 1 - 2.93iT - 11T^{2} \)
13 \( 1 + (-1.37 - 1.37i)T + 13iT^{2} \)
17 \( 1 + (5.24 - 5.24i)T - 17iT^{2} \)
19 \( 1 + (-2.39 + 2.39i)T - 19iT^{2} \)
23 \( 1 + (-4.68 - 4.68i)T + 23iT^{2} \)
29 \( 1 + (-3.81 + 3.81i)T - 29iT^{2} \)
31 \( 1 + (4.99 - 4.99i)T - 31iT^{2} \)
41 \( 1 - 5.64iT - 41T^{2} \)
43 \( 1 + (5.86 - 5.86i)T - 43iT^{2} \)
47 \( 1 - 8.48iT - 47T^{2} \)
53 \( 1 - 6.86iT - 53T^{2} \)
59 \( 1 + (6.32 - 6.32i)T - 59iT^{2} \)
61 \( 1 + (-8.73 + 8.73i)T - 61iT^{2} \)
67 \( 1 - 4.42iT - 67T^{2} \)
71 \( 1 + 9.41iT - 71T^{2} \)
73 \( 1 - 15.4iT - 73T^{2} \)
79 \( 1 + (-8.88 - 8.88i)T + 79iT^{2} \)
83 \( 1 + 13.9T + 83T^{2} \)
89 \( 1 + (4.91 + 4.91i)T + 89iT^{2} \)
97 \( 1 + (-6.59 + 6.59i)T - 97iT^{2} \)
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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

−10.22440771772858117852535790113, −9.563962534853235453206064617286, −8.816013630244991737400004020172, −7.914803895282368323810398969588, −7.01350102449220546531921556734, −6.27888219623664194185361903671, −4.61907030816857418683594608683, −3.94249043446651330137812504795, −2.71641575773646162579443041775, −1.35086612130126151366503099132, 0.59506020143249045363831643427, 2.09542037944229057643587029687, 3.12216197669054245167685324628, 5.10848225273438052012558214300, 5.75908957235627311391622852877, 6.70715080922469725435553563099, 7.38671585452750388744172820509, 8.586879839653123726797148021714, 8.771960632196927550235921936127, 9.708793501963331497631717306978

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