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Properties

Label 2-888-37.12-c1-0-15
Degree $2$
Conductor $888$
Sign $0.999 - 0.0253i$
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  + (0.766 + 0.642i)3-s + (3.36 − 1.22i)5-s + (4.08 − 1.48i)7-s + (0.173 + 0.984i)9-s + (0.923 + 1.59i)11-s + (−0.465 + 2.63i)13-s + (3.36 + 1.22i)15-s + (0.0676 + 0.383i)17-s + (−4.79 − 4.02i)19-s + (4.08 + 1.48i)21-s + (−2.17 + 3.76i)23-s + (5.98 − 5.02i)25-s + (−0.500 + 0.866i)27-s + (−4.52 − 7.84i)29-s − 10.1·31-s + ⋯
L(s)  = 1  + (0.442 + 0.371i)3-s + (1.50 − 0.547i)5-s + (1.54 − 0.561i)7-s + (0.0578 + 0.328i)9-s + (0.278 + 0.482i)11-s + (−0.128 + 0.731i)13-s + (0.868 + 0.316i)15-s + (0.0164 + 0.0930i)17-s + (−1.10 − 0.923i)19-s + (0.890 + 0.324i)21-s + (−0.452 + 0.784i)23-s + (1.19 − 1.00i)25-s + (−0.0962 + 0.166i)27-s + (−0.840 − 1.45i)29-s − 1.82·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.999 - 0.0253i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.999 - 0.0253i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.65849 + 0.0336782i\)
\(L(\frac12)\) \(\approx\) \(2.65849 + 0.0336782i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.766 - 0.642i)T \)
37 \( 1 + (-3.50 - 4.96i)T \)
good5 \( 1 + (-3.36 + 1.22i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-4.08 + 1.48i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-0.923 - 1.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.465 - 2.63i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.0676 - 0.383i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (4.79 + 4.02i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (2.17 - 3.76i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.52 + 7.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
41 \( 1 + (-0.650 + 3.69i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + (5.08 - 8.81i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.86 - 2.13i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (1.63 + 0.596i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.368 + 2.09i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-7.19 + 2.61i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-8.31 - 6.97i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 + (-7.01 + 2.55i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-2.15 - 12.2i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-9.11 - 3.31i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (3.25 - 5.63i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.889835476417950775433645120197, −9.395834692861006950477485445127, −8.593446971202131491745348468636, −7.73384213754903057790472508587, −6.69907168652311225383193144221, −5.56739449707929748598915382642, −4.76801667603002046220291708660, −4.05798240972212201837405581345, −2.14285725948800810259605667392, −1.64997619440468569556008087977, 1.70849242683877300927986610105, 2.19968010689741305331267518393, 3.55073493020196635174632120560, 5.10789789826777499508594111977, 5.74125116151774179360457398435, 6.60545916953370081882541417222, 7.69368575788695018315861884451, 8.527740840931779907726172123195, 9.113170905860513283024016341020, 10.23924273131490405972157709113

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