Properties

Label 2-847-11.3-c1-0-43
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.18 + 1.58i)2-s + (−0.867 − 2.66i)3-s + (1.64 − 5.04i)4-s + (0.360 + 0.261i)5-s + (6.13 + 4.46i)6-s + (0.309 − 0.951i)7-s + (2.76 + 8.50i)8-s + (−3.94 + 2.86i)9-s − 1.20·10-s − 14.8·12-s + (−0.364 + 0.265i)13-s + (0.835 + 2.57i)14-s + (0.386 − 1.18i)15-s + (−10.9 − 7.96i)16-s + (−3.90 − 2.83i)17-s + (4.07 − 12.5i)18-s + ⋯
L(s)  = 1  + (−1.54 + 1.12i)2-s + (−0.500 − 1.54i)3-s + (0.820 − 2.52i)4-s + (0.161 + 0.116i)5-s + (2.50 + 1.82i)6-s + (0.116 − 0.359i)7-s + (0.976 + 3.00i)8-s + (−1.31 + 0.956i)9-s − 0.380·10-s − 4.30·12-s + (−0.101 + 0.0735i)13-s + (0.223 + 0.687i)14-s + (0.0996 − 0.306i)15-s + (−2.74 − 1.99i)16-s + (−0.947 − 0.688i)17-s + (0.960 − 2.95i)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

\(L(1)\) \(\approx\) \(0.0119274 + 0.106765i\)
\(L(\frac12)\) \(\approx\) \(0.0119274 + 0.106765i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 - 1.58i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.867 + 2.66i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-0.360 - 0.261i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (0.364 - 0.265i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.90 + 2.83i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.335 + 1.03i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 4.57T + 23T^{2} \)
29 \( 1 + (0.613 - 1.88i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.68 - 4.85i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.26 + 6.95i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.547 - 1.68i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 + (-0.315 - 0.972i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.88 + 2.09i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.44 - 13.6i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.98 + 2.89i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 6.18T + 67T^{2} \)
71 \( 1 + (-4.79 - 3.48i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.512 - 1.57i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.91 + 2.12i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.74 - 6.35i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 5.21T + 89T^{2} \)
97 \( 1 + (-4.29 + 3.12i)T + (29.9 - 92.2i)T^{2} \)
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

−9.399660686567797316123575428726, −8.695636377293794960527189887480, −7.83926821075408554467919124377, −7.05940431457099959134521029403, −6.79264069964254595546140329042, −5.89057041973755322014256755823, −4.97302947168628427261292999987, −2.33753180529087762257487355049, −1.28538910638404859907114673898, −0.098087580831008067910494081516, 1.82399271158008973093450752573, 3.16143583100785863126398729849, 4.01317440718989665728604339604, 5.10408801146832155102252453588, 6.40581333938970891297195613819, 7.68701345278742351665040647582, 8.679318726601041797592490525572, 9.219142982429777080705374515399, 9.832246588601086656458665220670, 10.53053405472021168249724587979

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