Properties

Label 2-2842-1.1-c1-0-89
Degree $2$
Conductor $2842$
Sign $-1$
Analytic cond. $22.6934$
Root an. cond. $4.76376$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s − 2·9-s − 10-s − 3·11-s + 12-s + 13-s − 15-s + 16-s − 8·17-s − 2·18-s − 20-s − 3·22-s + 4·23-s + 24-s − 4·25-s + 26-s − 5·27-s − 29-s − 30-s + 3·31-s + 32-s − 3·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.904·11-s + 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s − 1.94·17-s − 0.471·18-s − 0.223·20-s − 0.639·22-s + 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.196·26-s − 0.962·27-s − 0.185·29-s − 0.182·30-s + 0.538·31-s + 0.176·32-s − 0.522·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2842\)    =    \(2 \cdot 7^{2} \cdot 29\)
Sign: $-1$
Analytic conductor: \(22.6934\)
Root analytic conductor: \(4.76376\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2842} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2842,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 \)
29 \( 1 + T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.237914033781111929312629804446, −7.81821112633712404756766747748, −6.76381712126143531253349126939, −6.16227613649540161988088922715, −5.11050011527156459020994869479, −4.49852693974322508959372476460, −3.49161543470772390818361140602, −2.79184388792227261783186696974, −1.93407659453531378223231825006, 0, 1.93407659453531378223231825006, 2.79184388792227261783186696974, 3.49161543470772390818361140602, 4.49852693974322508959372476460, 5.11050011527156459020994869479, 6.16227613649540161988088922715, 6.76381712126143531253349126939, 7.81821112633712404756766747748, 8.237914033781111929312629804446

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