↑

Properties

Label 2-847-11.9-c1-0-3
Degree $2$
Conductor $847$
Sign $0.569 + 0.821i$
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  + (−0.527 + 1.62i)2-s + (−1.85 + 1.34i)3-s + (−0.739 − 0.537i)4-s + (1.25 + 3.86i)5-s + (−1.20 − 3.72i)6-s + (−0.809 − 0.587i)7-s + (−1.49 + 1.08i)8-s + (0.695 − 2.14i)9-s − 6.94·10-s + 2.09·12-s + (1.01 − 3.11i)13-s + (1.38 − 1.00i)14-s + (−7.53 − 5.47i)15-s + (−1.54 − 4.74i)16-s + (−0.408 − 1.25i)17-s + (3.10 + 2.25i)18-s + ⋯
L(s)  = 1  + (−0.373 + 1.14i)2-s + (−1.07 + 0.777i)3-s + (−0.369 − 0.268i)4-s + (0.561 + 1.72i)5-s + (−0.493 − 1.51i)6-s + (−0.305 − 0.222i)7-s + (−0.530 + 0.385i)8-s + (0.231 − 0.713i)9-s − 2.19·10-s + 0.604·12-s + (0.280 − 0.864i)13-s + (0.369 − 0.268i)14-s + (−1.94 − 1.41i)15-s + (−0.385 − 1.18i)16-s + (−0.0989 − 0.304i)17-s + (0.732 + 0.532i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (372, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.569 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.381502 - 0.199697i\)
\(L(\frac12)\) \(\approx\) \(0.381502 - 0.199697i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (0.527 - 1.62i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (1.85 - 1.34i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-1.25 - 3.86i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-1.01 + 3.11i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.408 + 1.25i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.74 - 1.27i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 1.86T + 23T^{2} \)
29 \( 1 + (-0.197 - 0.143i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.12 - 6.53i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.206 + 0.150i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.63 + 3.36i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.01T + 43T^{2} \)
47 \( 1 + (-3.29 + 2.39i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.54 - 4.76i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.795 - 0.578i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.570 - 1.75i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 3.00T + 67T^{2} \)
71 \( 1 + (-2.00 - 6.15i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.80 - 5.67i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.71 - 5.26i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.639 - 1.96i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 16.6T + 89T^{2} \)
97 \( 1 + (0.798 - 2.45i)T + (-78.4 - 57.0i)T^{2} \)
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

−10.64871608353858614203010600787, −10.22795628850665397379990976906, −9.367495797057436713211965949244, −8.138928211129488153474818066048, −7.17973179473941673994219241330, −6.54836278228664380570109145777, −5.89405728710968228719891133883, −5.24329939108626263257615692626, −3.69530826782603343624911108866, −2.61539950369027199289609286810, 0.27555341394913068575234289006, 1.36396288889547340383453589967, 2.13313263266252753671109720313, 3.95843256689263784252028121459, 5.05005068353400131380094588469, 6.04370453390874349664861543988, 6.51213482764678882424187599857, 8.010778331686728892205664202575, 9.075174622355936138347163549401, 9.400344515353002135664449621416

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