↑

Properties

Label 2-847-11.3-c1-0-48
Degree $2$
Conductor $847$
Sign $-0.530 + 0.847i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.99 − 1.44i)2-s + (−0.5 − 1.53i)3-s + (1.26 − 3.88i)4-s + (2.80 + 2.03i)5-s + (−3.22 − 2.34i)6-s + (0.309 − 0.951i)7-s + (−1.58 − 4.89i)8-s + (0.309 − 0.224i)9-s + 8.55·10-s − 6.60·12-s + (−0.528 + 0.384i)13-s + (−0.762 − 2.34i)14-s + (1.73 − 5.33i)15-s + (−3.65 − 2.65i)16-s + (−0.919 − 0.668i)17-s + (0.291 − 0.896i)18-s + ⋯
L(s)  = 1  + (1.41 − 1.02i)2-s + (−0.288 − 0.888i)3-s + (0.631 − 1.94i)4-s + (1.25 + 0.911i)5-s + (−1.31 − 0.957i)6-s + (0.116 − 0.359i)7-s + (−0.561 − 1.72i)8-s + (0.103 − 0.0748i)9-s + 2.70·10-s − 1.90·12-s + (−0.146 + 0.106i)13-s + (−0.203 − 0.626i)14-s + (0.447 − 1.37i)15-s + (−0.913 − 0.663i)16-s + (−0.223 − 0.162i)17-s + (0.0686 − 0.211i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.530 + 0.847i$
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: \((2,\ 847,\ (\ :1/2),\ -0.530 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83081 - 3.30397i\)
\(L(\frac12)\) \(\approx\) \(1.83081 - 3.30397i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.99 + 1.44i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.5 + 1.53i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.80 - 2.03i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (0.528 - 0.384i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.919 + 0.668i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.87 - 5.77i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 6.66T + 23T^{2} \)
29 \( 1 + (-1.41 + 4.34i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.26 - 1.64i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.135 - 0.418i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.82 - 5.61i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 8.70T + 43T^{2} \)
47 \( 1 + (0.186 + 0.575i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.94 - 5.77i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.523 + 1.61i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.54 - 4.03i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 6.17T + 67T^{2} \)
71 \( 1 + (-4.38 - 3.18i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.07 - 6.37i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.14 + 1.55i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.41 - 3.93i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 0.698T + 89T^{2} \)
97 \( 1 + (-12.0 + 8.73i)T + (29.9 - 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.992580684453739502972224030107, −9.833139879757664686201836925839, −7.940918979587097006941009807756, −6.84239820498733193880003806364, −6.15034476709078957824655383551, −5.63309659491325224914445380262, −4.34668306540428838762996924790, −3.28896568302710147399689299736, −2.15987841174430643674823230411, −1.49716257668514713971247105643, 2.10085916120895844510161696509, 3.57340750659565673879329054482, 4.74279179580325859441895680121, 5.04888961015503415659317976587, 5.79894446055557591816337814768, 6.60579534548461141701362540497, 7.72196922722031961307396388674, 8.807078721435698016158180325231, 9.550319276276751975239402111637, 10.39559168693273375488745527458

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