Properties

Label 2-4680-1.1-c1-0-52
Degree $2$
Conductor $4680$
Sign $-1$
Analytic cond. $37.3699$
Root an. cond. $6.11309$
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  − 5-s + 2·7-s + 4.44·11-s − 13-s − 6.89·17-s + 0.449·19-s − 6.44·23-s + 25-s − 4·29-s − 4.44·31-s − 2·35-s + 4.89·37-s − 10.8·41-s + 11.3·43-s + 2·47-s − 3·49-s − 1.10·53-s − 4.44·55-s − 9.34·59-s − 5.79·61-s + 65-s − 5.10·67-s + 3.55·71-s − 14.6·73-s + 8.89·77-s + 4.89·79-s − 2·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.755·7-s + 1.34·11-s − 0.277·13-s − 1.67·17-s + 0.103·19-s − 1.34·23-s + 0.200·25-s − 0.742·29-s − 0.799·31-s − 0.338·35-s + 0.805·37-s − 1.70·41-s + 1.73·43-s + 0.291·47-s − 0.428·49-s − 0.151·53-s − 0.599·55-s − 1.21·59-s − 0.742·61-s + 0.124·65-s − 0.623·67-s + 0.421·71-s − 1.72·73-s + 1.01·77-s + 0.551·79-s − 0.219·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4680\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(37.3699\)
Root analytic conductor: \(6.11309\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4680,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
13 \( 1 + T \)
good7 \( 1 - 2T + 7T^{2} \)
11 \( 1 - 4.44T + 11T^{2} \)
17 \( 1 + 6.89T + 17T^{2} \)
19 \( 1 - 0.449T + 19T^{2} \)
23 \( 1 + 6.44T + 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + 4.44T + 31T^{2} \)
37 \( 1 - 4.89T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 + 1.10T + 53T^{2} \)
59 \( 1 + 9.34T + 59T^{2} \)
61 \( 1 + 5.79T + 61T^{2} \)
67 \( 1 + 5.10T + 67T^{2} \)
71 \( 1 - 3.55T + 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 - 4.89T + 79T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 7.79T + 97T^{2} \)
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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

−7.84751013594059444218594388666, −7.33469462211241109564909565564, −6.47573440382732111789257703802, −5.87746010886331827597204936922, −4.72854018272802603278693727786, −4.26837272076479963526390853975, −3.51915981516653765988320901798, −2.24704576644844845324494842483, −1.49108367490569580390339329579, 0, 1.49108367490569580390339329579, 2.24704576644844845324494842483, 3.51915981516653765988320901798, 4.26837272076479963526390853975, 4.72854018272802603278693727786, 5.87746010886331827597204936922, 6.47573440382732111789257703802, 7.33469462211241109564909565564, 7.84751013594059444218594388666

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