Properties

Label 2-2842-1.1-c1-0-90
Degree $2$
Conductor $2842$
Sign $-1$
Analytic cond. $22.6934$
Root an. cond. $4.76376$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 1.41·5-s + 8-s − 3·9-s + 1.41·10-s − 2·11-s − 7.07·13-s + 16-s + 1.41·17-s − 3·18-s + 2.82·19-s + 1.41·20-s − 2·22-s − 6·23-s − 2.99·25-s − 7.07·26-s − 29-s − 1.41·31-s + 32-s + 1.41·34-s − 3·36-s − 4·37-s + 2.82·38-s + 1.41·40-s + 7.07·41-s − 4·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.632·5-s + 0.353·8-s − 9-s + 0.447·10-s − 0.603·11-s − 1.96·13-s + 0.250·16-s + 0.342·17-s − 0.707·18-s + 0.648·19-s + 0.316·20-s − 0.426·22-s − 1.25·23-s − 0.599·25-s − 1.38·26-s − 0.185·29-s − 0.254·31-s + 0.176·32-s + 0.242·34-s − 0.5·36-s − 0.657·37-s + 0.458·38-s + 0.223·40-s + 1.10·41-s − 0.609·43-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: no
Arithmetic: yes
Character: Trivial
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 \( 1 + 3T^{2} \)
5 \( 1 - 1.41T + 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 7.07T + 13T^{2} \)
17 \( 1 - 1.41T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
31 \( 1 + 1.41T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 - 7.07T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 9.89T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 - 7.07T + 89T^{2} \)
97 \( 1 + 4.24T + 97T^{2} \)
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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

−8.134192960944930627699696299254, −7.69206300215806924993208924290, −6.77477756733577158503456457859, −5.85590396186397951401981424018, −5.34721497262056777570880583713, −4.69835287911109903932785863065, −3.48091181158175284480969721410, −2.62397009510143994678621541863, −1.94548932682159096552640825297, 0, 1.94548932682159096552640825297, 2.62397009510143994678621541863, 3.48091181158175284480969721410, 4.69835287911109903932785863065, 5.34721497262056777570880583713, 5.85590396186397951401981424018, 6.77477756733577158503456457859, 7.69206300215806924993208924290, 8.134192960944930627699696299254

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