Properties

Label 2-6840-1.1-c1-0-22
Degree $2$
Conductor $6840$
Sign $1$
Analytic cond. $54.6176$
Root an. cond. $7.39037$
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  − 5-s + 3.20·7-s − 5.20·11-s + 5.45·13-s − 4·17-s + 19-s + 8.65·23-s + 25-s + 3.20·29-s − 2·31-s − 3.20·35-s − 11.8·37-s + 9.85·41-s − 3.45·43-s − 8.65·47-s + 3.25·49-s + 10.6·53-s + 5.20·55-s + 13.0·59-s + 3.74·61-s − 5.45·65-s + 4·67-s − 6.90·71-s + 6·73-s − 16.6·77-s − 6.15·79-s + 8.40·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.21·7-s − 1.56·11-s + 1.51·13-s − 0.970·17-s + 0.229·19-s + 1.80·23-s + 0.200·25-s + 0.594·29-s − 0.359·31-s − 0.541·35-s − 1.94·37-s + 1.53·41-s − 0.526·43-s − 1.26·47-s + 0.464·49-s + 1.46·53-s + 0.701·55-s + 1.70·59-s + 0.479·61-s − 0.676·65-s + 0.488·67-s − 0.819·71-s + 0.702·73-s − 1.89·77-s − 0.692·79-s + 0.922·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6840\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(54.6176\)
Root analytic conductor: \(7.39037\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6840} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.041966435\)
\(L(\frac12)\) \(\approx\) \(2.041966435\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
19 \( 1 - T \)
good7 \( 1 - 3.20T + 7T^{2} \)
11 \( 1 + 5.20T + 11T^{2} \)
13 \( 1 - 5.45T + 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
23 \( 1 - 8.65T + 23T^{2} \)
29 \( 1 - 3.20T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 11.8T + 37T^{2} \)
41 \( 1 - 9.85T + 41T^{2} \)
43 \( 1 + 3.45T + 43T^{2} \)
47 \( 1 + 8.65T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 - 13.0T + 59T^{2} \)
61 \( 1 - 3.74T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 6.90T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 6.15T + 79T^{2} \)
83 \( 1 - 8.40T + 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 - 0.142T + 97T^{2} \)
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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

−8.188199105191009634253845765490, −7.24610401641665183509485782090, −6.78150661206004317132619650485, −5.62348746928534938394869660433, −5.13611453727594716769207104007, −4.48819527642475406629501672974, −3.58284454442487163696595953688, −2.74324356271774964766129324211, −1.78396171876200716098222588106, −0.74834249765321032478226912845, 0.74834249765321032478226912845, 1.78396171876200716098222588106, 2.74324356271774964766129324211, 3.58284454442487163696595953688, 4.48819527642475406629501672974, 5.13611453727594716769207104007, 5.62348746928534938394869660433, 6.78150661206004317132619650485, 7.24610401641665183509485782090, 8.188199105191009634253845765490

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