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Properties

Label 2-847-11.3-c1-0-47
Degree $2$
Conductor $847$
Sign $-0.975 - 0.220i$
Analytic cond. $6.76332$
Root an. cond. $2.60064$
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  + (0.666 − 0.484i)2-s + (−0.296 − 0.913i)3-s + (−0.408 + 1.25i)4-s + (−2.41 − 1.75i)5-s + (−0.640 − 0.465i)6-s + (0.309 − 0.951i)7-s + (0.845 + 2.60i)8-s + (1.68 − 1.22i)9-s − 2.45·10-s + 1.26·12-s + (−1.78 + 1.29i)13-s + (−0.254 − 0.783i)14-s + (−0.886 + 2.72i)15-s + (−0.315 − 0.229i)16-s + (−3.56 − 2.59i)17-s + (0.528 − 1.62i)18-s + ⋯
L(s)  = 1  + (0.471 − 0.342i)2-s + (−0.171 − 0.527i)3-s + (−0.204 + 0.628i)4-s + (−1.08 − 0.784i)5-s + (−0.261 − 0.189i)6-s + (0.116 − 0.359i)7-s + (0.298 + 0.919i)8-s + (0.560 − 0.406i)9-s − 0.777·10-s + 0.366·12-s + (−0.493 + 0.358i)13-s + (−0.0680 − 0.209i)14-s + (−0.228 + 0.704i)15-s + (−0.0789 − 0.0573i)16-s + (−0.864 − 0.628i)17-s + (0.124 − 0.383i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.975 - 0.220i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0508593 + 0.455254i\)
\(L(\frac12)\) \(\approx\) \(0.0508593 + 0.455254i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-0.666 + 0.484i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.296 + 0.913i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.41 + 1.75i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (1.78 - 1.29i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.56 + 2.59i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.533 + 1.64i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 8.39T + 23T^{2} \)
29 \( 1 + (-1.01 + 3.13i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.04 - 4.39i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.71 - 8.35i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.66 - 5.12i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 9.44T + 43T^{2} \)
47 \( 1 + (1.66 + 5.13i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.60 - 5.52i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.07 + 3.30i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (10.5 + 7.63i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 4.32T + 67T^{2} \)
71 \( 1 + (3.56 + 2.58i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.55 + 14.0i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.81 + 4.22i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.17 + 4.48i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + (2.30 - 1.67i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.635292317051491351119748604307, −8.766302286243030083257390330094, −7.86732274946003485588890077384, −7.40316831846485856493455882450, −6.35099140267416060364395104669, −4.74950278375731590592595926747, −4.39235941798698809057967449079, −3.43162806122065683366194806647, −1.92922098397732725711865609906, −0.18877148246218600614129135999, 2.09611179648178777506138733233, 3.83583274346183784309146367249, 4.21822855575735012877801798005, 5.35541561298852613063610182831, 6.15748310919234353209224400961, 7.24053115235018442017568577442, 7.82248708426579703127704799734, 9.066709933872312872895545917974, 9.980427293289407203441719540066, 10.71543028524842372101771272750

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