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Properties

Label 2-888-37.10-c1-0-10
Degree $2$
Conductor $888$
Sign $0.629 + 0.776i$
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 + (−0.0695 − 0.120i)5-s + (0.199 + 0.344i)7-s + (−0.499 + 0.866i)9-s + 3.78·11-s + (−0.893 − 1.54i)13-s + (−0.0695 + 0.120i)15-s + (0.5 − 0.866i)17-s + (0.967 + 1.67i)19-s + (0.199 − 0.344i)21-s + 1.53·23-s + (2.49 − 4.31i)25-s + 0.999·27-s − 2.72·29-s + 4.92·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.0311 − 0.0539i)5-s + (0.0752 + 0.130i)7-s + (−0.166 + 0.288i)9-s + 1.14·11-s + (−0.247 − 0.428i)13-s + (−0.0179 + 0.0311i)15-s + (0.121 − 0.210i)17-s + (0.222 + 0.384i)19-s + (0.0434 − 0.0752i)21-s + 0.320·23-s + (0.498 − 0.862i)25-s + 0.192·27-s − 0.505·29-s + 0.884·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.629 + 0.776i$
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: \((2,\ 888,\ (\ :1/2),\ 0.629 + 0.776i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32957 - 0.633625i\)
\(L(\frac12)\) \(\approx\) \(1.32957 - 0.633625i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.08 + 0.0647i)T \)
good5 \( 1 + (0.0695 + 0.120i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.199 - 0.344i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.78T + 11T^{2} \)
13 \( 1 + (0.893 + 1.54i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.967 - 1.67i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.53T + 23T^{2} \)
29 \( 1 + 2.72T + 29T^{2} \)
31 \( 1 - 4.92T + 31T^{2} \)
41 \( 1 + (5.77 + 10.0i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 7.22T + 43T^{2} \)
47 \( 1 + 1.32T + 47T^{2} \)
53 \( 1 + (-0.338 + 0.585i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.49 + 2.58i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.48 + 6.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.62 - 6.28i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.24 + 7.35i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.149T + 73T^{2} \)
79 \( 1 + (-0.952 - 1.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.925 + 1.60i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.08 - 8.81i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−10.02600590000060011496700030745, −9.122860752377568230725265295408, −8.305366983651936639621316177970, −7.38912202010497022372277515339, −6.57767620048521518646848377285, −5.75217610517148497030071625881, −4.73806684653641264364659654318, −3.60361822077170046845808179227, −2.28662652110126577990155150824, −0.892265099160117569649908696293, 1.25656852633820591506467553250, 2.92355763999627753147206453747, 4.05309762642077659148834512358, 4.80384524174888323713087671121, 5.94096305927223675748742666119, 6.74502144260577549409891396798, 7.63995708281230954793258041960, 8.821922309393610672113155279089, 9.400774937246969055420373553271, 10.18399746276757183857858416598

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