Properties

Label 2-2842-1.1-c1-0-77
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

Related objects

Downloads

Learn more

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: \(1\)
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 \( 1 + 2.40T + 3T^{2} \)
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 \( 1 + 3.23T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.476975119599703023820426059951, −7.01449603554066323796229114943, −6.82651662222004008508547463027, −5.94432127309126770530840730405, −5.21487319081941694376704900604, −4.94637061191778016761641610990, −3.78533211784464000497506071189, −2.51790914657472791020462889923, −1.62230202159274976889338111277, 0, 1.62230202159274976889338111277, 2.51790914657472791020462889923, 3.78533211784464000497506071189, 4.94637061191778016761641610990, 5.21487319081941694376704900604, 5.94432127309126770530840730405, 6.82651662222004008508547463027, 7.01449603554066323796229114943, 8.476975119599703023820426059951

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