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Properties

Label 2-888-37.26-c1-0-1
Degree $2$
Conductor $888$
Sign $-0.814 + 0.580i$
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 + (−2.05 + 3.56i)5-s + (−2.29 + 3.97i)7-s + (−0.499 − 0.866i)9-s + 1.83·11-s + (0.0846 − 0.146i)13-s + (−2.05 − 3.56i)15-s + (0.5 + 0.866i)17-s + (−2.03 + 3.51i)19-s + (−2.29 − 3.97i)21-s + 0.527·23-s + (−5.96 − 10.3i)25-s + 0.999·27-s + 5.22·29-s + 6.94·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.920 + 1.59i)5-s + (−0.867 + 1.50i)7-s + (−0.166 − 0.288i)9-s + 0.551·11-s + (0.0234 − 0.0406i)13-s + (−0.531 − 0.920i)15-s + (0.121 + 0.210i)17-s + (−0.465 + 0.806i)19-s + (−0.500 − 0.867i)21-s + 0.109·23-s + (−1.19 − 2.06i)25-s + 0.192·27-s + 0.971·29-s + 1.24·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.814 + 0.580i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ -0.814 + 0.580i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.200534 - 0.626663i\)
\(L(\frac12)\) \(\approx\) \(0.200534 - 0.626663i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.84 + 1.68i)T \)
good5 \( 1 + (2.05 - 3.56i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.29 - 3.97i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 1.83T + 11T^{2} \)
13 \( 1 + (-0.0846 + 0.146i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.03 - 3.51i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.527T + 23T^{2} \)
29 \( 1 - 5.22T + 29T^{2} \)
31 \( 1 - 6.94T + 31T^{2} \)
41 \( 1 + (-4.63 + 8.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 - 1.64T + 47T^{2} \)
53 \( 1 + (-1.82 - 3.15i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.50 - 9.53i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.963 + 1.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.851 - 1.47i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.30 - 5.72i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.89T + 73T^{2} \)
79 \( 1 + (6.49 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.94 - 5.10i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.62 + 9.74i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−10.42755587635772736361571401376, −10.09882639819232805256564219446, −8.932243962820556929839136995059, −8.224528996608244224090106006655, −6.97212269177330172404886967400, −6.37612698833721071468026027498, −5.61779631194330983615617085867, −4.13510600588180049964624044601, −3.29419684563654373188724205973, −2.49704648241644259740919542196, 0.36844509248436165341005512074, 1.22772634270977437711403151302, 3.32751819914484237480012741642, 4.35102614845750616385698217285, 4.90592895144130401539083726473, 6.38784916093010059214231387834, 7.03759701276914770924465218822, 7.960536059187097470927845861425, 8.649572398972774828482091902780, 9.590486955896984589114387512432

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