↑

Properties

Label 2-847-11.5-c1-0-1
Degree $2$
Conductor $847$
Sign $0.437 - 0.899i$
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  + (−0.309 − 0.951i)2-s + (−1.61 − 1.17i)3-s + (0.809 − 0.587i)4-s + (−0.618 + 1.90i)5-s + (−0.618 + 1.90i)6-s + (−0.809 + 0.587i)7-s + (−2.42 − 1.76i)8-s + (0.309 + 0.951i)9-s + 1.99·10-s − 2·12-s + (−1.23 − 3.80i)13-s + (0.809 + 0.587i)14-s + (3.23 − 2.35i)15-s + (−0.309 + 0.951i)16-s + (−1.23 + 3.80i)17-s + (0.809 − 0.587i)18-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.934 − 0.678i)3-s + (0.404 − 0.293i)4-s + (−0.276 + 0.850i)5-s + (−0.252 + 0.776i)6-s + (−0.305 + 0.222i)7-s + (−0.858 − 0.623i)8-s + (0.103 + 0.317i)9-s + 0.632·10-s − 0.577·12-s + (−0.342 − 1.05i)13-s + (0.216 + 0.157i)14-s + (0.835 − 0.607i)15-s + (−0.0772 + 0.237i)16-s + (−0.299 + 0.922i)17-s + (0.190 − 0.138i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.437 - 0.899i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.437 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.205677 + 0.128601i\)
\(L(\frac12)\) \(\approx\) \(0.205677 + 0.128601i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.61 + 1.17i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.618 - 1.90i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (1.23 + 3.80i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.23 - 3.80i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + (4.85 - 3.52i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.09 - 9.51i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-4.85 + 3.52i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.23 - 2.35i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + (-8.09 - 5.87i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.85 + 5.70i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.61 - 1.17i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + (3.70 - 11.4i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (6.47 - 4.70i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.47 + 7.60i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (3.09 + 9.51i)T + (-78.4 + 57.0i)T^{2} \)
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

−10.53783292252259672799573967687, −9.925488317809533592284781535941, −8.744382116863217921194587802858, −7.53049904643247804316144323590, −6.75684787467917635644124362813, −6.14080678657937278241661702766, −5.35594289706338941167640462441, −3.59008820230332564870684767909, −2.70022992069080578136801578278, −1.37078141799585361891768254651, 0.14191867392086788064135270729, 2.36046672107023033874017675541, 4.02234352457002809873634029249, 4.75069485523418102033865767277, 5.74407884764495685934017929081, 6.48738697640217868571427883917, 7.45534927222808867924067455007, 8.246538530213878300824772846803, 9.250601707180879011440518939686, 9.868343253688757251112023056616

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