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Properties

Label 2-847-11.9-c1-0-29
Degree $2$
Conductor $847$
Sign $0.780 + 0.625i$
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  + (0.309 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.780 + 0.625i$
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: \((2,\ 847,\ (\ :1/2),\ 0.780 + 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43471 - 0.504177i\)
\(L(\frac12)\) \(\approx\) \(1.43471 - 0.504177i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.09 - 9.51i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.54173452826358127848889082849, −9.564432774291387181402221063532, −8.238408421815532229444133342894, −7.85354931613826442986524469310, −6.31627276940851103054417792078, −5.55093024028473811839734572095, −4.55078410350496583271100922942, −3.90389022531421251169546721583, −2.52829665064565133915829300181, −0.969623683875399200582380364888, 1.19143020140226961146844775396, 2.60198949879693018158301370058, 4.19347482947790217823711413907, 5.30317302911706343136487690206, 6.14439516316244049904216342482, 6.85954001395315968840633825254, 7.21659241526536338659973143156, 8.205195743573113201695215012334, 9.519938145045538372403832266560, 10.66757762101968191586694264788

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