Properties

Label 2-888-296.43-c1-0-43
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: $\chi_{888} (43, \cdot )$
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

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

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