Properties

Label 2-2842-1.1-c1-0-40
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2.40·3-s + 4-s − 2.20·5-s + 2.40·6-s + 8-s + 2.79·9-s − 2.20·10-s − 1.18·11-s + 2.40·12-s + 1.24·13-s − 5.31·15-s + 16-s + 5.96·17-s + 2.79·18-s + 2.19·19-s − 2.20·20-s − 1.18·22-s + 7.74·23-s + 2.40·24-s − 0.125·25-s + 1.24·26-s − 0.497·27-s − 29-s − 5.31·30-s + 9.31·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.38·3-s + 0.5·4-s − 0.987·5-s + 0.982·6-s + 0.353·8-s + 0.931·9-s − 0.698·10-s − 0.358·11-s + 0.694·12-s + 0.344·13-s − 1.37·15-s + 0.250·16-s + 1.44·17-s + 0.658·18-s + 0.503·19-s − 0.493·20-s − 0.253·22-s + 1.61·23-s + 0.491·24-s − 0.0250·25-s + 0.243·26-s − 0.0956·27-s − 0.185·29-s − 0.970·30-s + 1.67·31-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: \(0\)
Selberg data: \((2,\ 2842,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.263037497\)
\(L(\frac12)\) \(\approx\) \(4.263037497\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.40T + 3T^{2} \)
5 \( 1 + 2.20T + 5T^{2} \)
11 \( 1 + 1.18T + 11T^{2} \)
13 \( 1 - 1.24T + 13T^{2} \)
17 \( 1 - 5.96T + 17T^{2} \)
19 \( 1 - 2.19T + 19T^{2} \)
23 \( 1 - 7.74T + 23T^{2} \)
31 \( 1 - 9.31T + 31T^{2} \)
37 \( 1 + 6.90T + 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + 5.76T + 43T^{2} \)
47 \( 1 + 5.82T + 47T^{2} \)
53 \( 1 - 1.68T + 53T^{2} \)
59 \( 1 + 7.97T + 59T^{2} \)
61 \( 1 - 8.11T + 61T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 - 4.61T + 71T^{2} \)
73 \( 1 - 0.0394T + 73T^{2} \)
79 \( 1 + 2.42T + 79T^{2} \)
83 \( 1 - 5.46T + 83T^{2} \)
89 \( 1 + 7.26T + 89T^{2} \)
97 \( 1 - 3.23T + 97T^{2} \)
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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

−8.531264035881866987204464976793, −7.977322256020306115142058995191, −7.49230941532195937429480600804, −6.68083128926402428858037913056, −5.51873412781968226757053157172, −4.72938638812503851432820357597, −3.70373708554858429577736277636, −3.29100945717872858758067622425, −2.54273354544628899250003261932, −1.16073713084145760465515114173, 1.16073713084145760465515114173, 2.54273354544628899250003261932, 3.29100945717872858758067622425, 3.70373708554858429577736277636, 4.72938638812503851432820357597, 5.51873412781968226757053157172, 6.68083128926402428858037913056, 7.49230941532195937429480600804, 7.977322256020306115142058995191, 8.531264035881866987204464976793

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