Properties

Label 2-2842-1.1-c1-0-88
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 + 2.42·3-s + 4-s − 0.260·5-s − 2.42·6-s − 8-s + 2.88·9-s + 0.260·10-s + 2.14·11-s + 2.42·12-s − 6.34·13-s − 0.632·15-s + 16-s − 2.22·17-s − 2.88·18-s − 8.43·19-s − 0.260·20-s − 2.14·22-s + 4.38·23-s − 2.42·24-s − 4.93·25-s + 6.34·26-s − 0.274·27-s − 29-s + 0.632·30-s + 3.04·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.40·3-s + 0.5·4-s − 0.116·5-s − 0.990·6-s − 0.353·8-s + 0.962·9-s + 0.0824·10-s + 0.647·11-s + 0.700·12-s − 1.75·13-s − 0.163·15-s + 0.250·16-s − 0.540·17-s − 0.680·18-s − 1.93·19-s − 0.0583·20-s − 0.457·22-s + 0.914·23-s − 0.495·24-s − 0.986·25-s + 1.24·26-s − 0.0528·27-s − 0.185·29-s + 0.115·30-s + 0.546·31-s − 0.176·32-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 \( 1 - 2.42T + 3T^{2} \)
5 \( 1 + 0.260T + 5T^{2} \)
11 \( 1 - 2.14T + 11T^{2} \)
13 \( 1 + 6.34T + 13T^{2} \)
17 \( 1 + 2.22T + 17T^{2} \)
19 \( 1 + 8.43T + 19T^{2} \)
23 \( 1 - 4.38T + 23T^{2} \)
31 \( 1 - 3.04T + 31T^{2} \)
37 \( 1 + 7.24T + 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 - 0.195T + 43T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 + 0.390T + 53T^{2} \)
59 \( 1 - 14.7T + 59T^{2} \)
61 \( 1 - 9.83T + 61T^{2} \)
67 \( 1 - 9.53T + 67T^{2} \)
71 \( 1 - 2.18T + 71T^{2} \)
73 \( 1 + 4.18T + 73T^{2} \)
79 \( 1 + 4.48T + 79T^{2} \)
83 \( 1 + 1.91T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 - 0.283T + 97T^{2} \)
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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

−8.553956315096583971101724862507, −7.912616983592737138072445810780, −6.98092775270042140545544736576, −6.63914454975993484410180241748, −5.20948619483113184034008908366, −4.24888724408091617151288131672, −3.38019591522368905582936580733, −2.37366976238035824048389540655, −1.86890974327642007631453148872, 0, 1.86890974327642007631453148872, 2.37366976238035824048389540655, 3.38019591522368905582936580733, 4.24888724408091617151288131672, 5.20948619483113184034008908366, 6.63914454975993484410180241748, 6.98092775270042140545544736576, 7.912616983592737138072445810780, 8.553956315096583971101724862507

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