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Properties

Label 2-888-37.10-c1-0-1
Degree $2$
Conductor $888$
Sign $0.895 - 0.445i$
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  + (−0.5 − 0.866i)3-s + (−0.872 − 1.51i)5-s + (1.09 + 1.89i)7-s + (−0.499 + 0.866i)9-s − 4.61·11-s + (3.30 + 5.73i)13-s + (−0.872 + 1.51i)15-s + (0.5 − 0.866i)17-s + (3.56 + 6.17i)19-s + (1.09 − 1.89i)21-s + 4.93·23-s + (0.975 − 1.69i)25-s + 0.999·27-s + 0.491·29-s − 1.87·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.390 − 0.676i)5-s + (0.413 + 0.716i)7-s + (−0.166 + 0.288i)9-s − 1.39·11-s + (0.917 + 1.58i)13-s + (−0.225 + 0.390i)15-s + (0.121 − 0.210i)17-s + (0.817 + 1.41i)19-s + (0.238 − 0.413i)21-s + 1.02·23-s + (0.195 − 0.338i)25-s + 0.192·27-s + 0.0913·29-s − 0.336·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.895 - 0.445i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.895 - 0.445i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19993 + 0.282007i\)
\(L(\frac12)\) \(\approx\) \(1.19993 + 0.282007i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-1.26 + 5.95i)T \)
good5 \( 1 + (0.872 + 1.51i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.09 - 1.89i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.61T + 11T^{2} \)
13 \( 1 + (-3.30 - 5.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.56 - 6.17i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.93T + 23T^{2} \)
29 \( 1 - 0.491T + 29T^{2} \)
31 \( 1 + 1.87T + 31T^{2} \)
41 \( 1 + (-4.14 - 7.17i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 7.60T + 43T^{2} \)
47 \( 1 - 3.68T + 47T^{2} \)
53 \( 1 + (-2.84 + 4.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.49 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.02 - 6.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.72 - 6.44i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.55 - 13.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.7T + 73T^{2} \)
79 \( 1 + (3.95 + 6.85i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.87 - 10.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.21 + 7.29i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.38T + 97T^{2} \)
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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

−10.24161697411136248411422243702, −9.147406096061206273276831561307, −8.435868183249807834268043767157, −7.78637055470967467775904946890, −6.79560318261282300652505224027, −5.67868665918484411224229091625, −5.07919715553025768287825794319, −3.95326817319987877986038630577, −2.48343589478493301465091938661, −1.25799541532834659750056024548, 0.71218923006885170924793780413, 2.87402625398870310715197236017, 3.51884986868490161928803798729, 4.90490464559849943491999678164, 5.44752175149271020257087566469, 6.73232059800348853939367756914, 7.59663853495567241030140893385, 8.194913845274815207117307972940, 9.343114405995276971037863384425, 10.41519207996988736002549972501

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