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Properties

Label 2-888-37.10-c1-0-12
Degree $2$
Conductor $888$
Sign $0.577 + 0.816i$
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 + (−0.719 − 1.24i)5-s + (−0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s − 0.950·11-s + (−0.219 − 0.380i)13-s + (0.719 − 1.24i)15-s + (3.19 − 5.53i)17-s + (−1.91 − 3.31i)19-s + (0.499 − 0.866i)21-s + 8.21·23-s + (1.46 − 2.53i)25-s − 0.999·27-s + 6.48·29-s − 3.38·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.321 − 0.557i)5-s + (−0.188 − 0.327i)7-s + (−0.166 + 0.288i)9-s − 0.286·11-s + (−0.0609 − 0.105i)13-s + (0.185 − 0.321i)15-s + (0.774 − 1.34i)17-s + (−0.439 − 0.760i)19-s + (0.109 − 0.188i)21-s + 1.71·23-s + (0.292 − 0.507i)25-s − 0.192·27-s + 1.20·29-s − 0.608·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.577 + 0.816i$
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: \((2,\ 888,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27015 - 0.657699i\)
\(L(\frac12)\) \(\approx\) \(1.27015 - 0.657699i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.07 + 0.337i)T \)
good5 \( 1 + (0.719 + 1.24i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.950T + 11T^{2} \)
13 \( 1 + (0.219 + 0.380i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.19 + 5.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.91 + 3.31i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.21T + 23T^{2} \)
29 \( 1 - 6.48T + 29T^{2} \)
31 \( 1 + 3.38T + 31T^{2} \)
41 \( 1 + (3.86 + 6.69i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 2.43T + 43T^{2} \)
47 \( 1 + 4.48T + 47T^{2} \)
53 \( 1 + (-1.63 + 2.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.964 - 1.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0497 + 0.0861i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.46 + 4.26i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.87 - 13.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.29T + 73T^{2} \)
79 \( 1 + (-6.80 - 11.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.26 + 10.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.82 + 10.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.80T + 97T^{2} \)
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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

−10.01332021907355810061562863242, −9.041681012355431119847047564889, −8.553281028930934967889515199703, −7.44652486379122660936709448078, −6.75379148641797264605883124768, −5.21865594539263438221554855314, −4.80574454784189914857365639373, −3.57647649729623264105486191679, −2.62561853636268473089299749610, −0.70895833316278859234865308958, 1.48533697533442936403604035824, 2.87388089446630718648926277317, 3.63486774224914245318815646931, 5.02392089385633544852560443850, 6.10820285318318198060929603563, 6.85295077858916450415043343353, 7.75164234833241826020102811372, 8.460785258189643775994323941310, 9.315824408136144814594645977301, 10.41874404691406746447084993847

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