Properties

Label 2-6840-1.1-c1-0-42
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 + 5.23·7-s + 1.23·11-s + 0.763·13-s − 4.47·17-s + 19-s − 2.47·23-s + 25-s − 0.763·29-s − 8.94·31-s + 5.23·35-s + 3.23·37-s + 9.70·41-s + 5.23·43-s − 2.47·47-s + 20.4·49-s + 0.472·53-s + 1.23·55-s + 10.4·59-s + 4.47·61-s + 0.763·65-s + 1.52·67-s + 12.4·73-s + 6.47·77-s − 4·83-s − 4.47·85-s + 1.70·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.97·7-s + 0.372·11-s + 0.211·13-s − 1.08·17-s + 0.229·19-s − 0.515·23-s + 0.200·25-s − 0.141·29-s − 1.60·31-s + 0.885·35-s + 0.532·37-s + 1.51·41-s + 0.798·43-s − 0.360·47-s + 2.91·49-s + 0.0648·53-s + 0.166·55-s + 1.36·59-s + 0.572·61-s + 0.0947·65-s + 0.186·67-s + 1.45·73-s + 0.737·77-s − 0.439·83-s − 0.485·85-s + 0.181·89-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

\(L(1)\) \(\approx\) \(3.029231967\)
\(L(\frac12)\) \(\approx\) \(3.029231967\)
\(L(\frac{3}{2})\) 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 \( 1 - 5.23T + 7T^{2} \)
11 \( 1 - 1.23T + 11T^{2} \)
13 \( 1 - 0.763T + 13T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 + 0.763T + 29T^{2} \)
31 \( 1 + 8.94T + 31T^{2} \)
37 \( 1 - 3.23T + 37T^{2} \)
41 \( 1 - 9.70T + 41T^{2} \)
43 \( 1 - 5.23T + 43T^{2} \)
47 \( 1 + 2.47T + 47T^{2} \)
53 \( 1 - 0.472T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 - 4.47T + 61T^{2} \)
67 \( 1 - 1.52T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 - 1.70T + 89T^{2} \)
97 \( 1 + 4.76T + 97T^{2} \)
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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

−7.87990044780814652222378503137, −7.44690062104442250101359635298, −6.56366173577286990451969727169, −5.71511838170767434762902238401, −5.17005435552545685518930338771, −4.38084393641928110390797943220, −3.82515413829852694352333434621, −2.38412167292024674129098985164, −1.89776623332570026488386716960, −0.939480966635311620287737337731, 0.939480966635311620287737337731, 1.89776623332570026488386716960, 2.38412167292024674129098985164, 3.82515413829852694352333434621, 4.38084393641928110390797943220, 5.17005435552545685518930338771, 5.71511838170767434762902238401, 6.56366173577286990451969727169, 7.44690062104442250101359635298, 7.87990044780814652222378503137

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