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Properties

Label 2-888-37.34-c1-0-16
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: $\chi_{888} (145, \cdot )$
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

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

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