Properties

Label 2-4680-1.1-c1-0-50
Degree $2$
Conductor $4680$
Sign $-1$
Analytic cond. $37.3699$
Root an. cond. $6.11309$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 4·11-s + 13-s + 2·17-s − 4·19-s − 4·23-s + 25-s + 6·29-s − 4·31-s + 6·37-s + 2·41-s + 12·43-s − 8·47-s − 7·49-s − 14·53-s − 4·55-s − 12·59-s − 2·61-s + 65-s − 8·71-s − 2·73-s + 8·79-s − 12·83-s + 2·85-s − 6·89-s − 4·95-s − 10·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.20·11-s + 0.277·13-s + 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s + 0.986·37-s + 0.312·41-s + 1.82·43-s − 1.16·47-s − 49-s − 1.92·53-s − 0.539·55-s − 1.56·59-s − 0.256·61-s + 0.124·65-s − 0.949·71-s − 0.234·73-s + 0.900·79-s − 1.31·83-s + 0.216·85-s − 0.635·89-s − 0.410·95-s − 1.01·97-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: yes
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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−7.984189477416458754672061917520, −7.34689845150244954575088779748, −6.23186765922244566574282930469, −5.94629015825883792891429139086, −4.94284031943159442030500635338, −4.32039212045326519058636437187, −3.17969977983531980383290806892, −2.46257396202073637007313610811, −1.45821482974105581536616908885, 0, 1.45821482974105581536616908885, 2.46257396202073637007313610811, 3.17969977983531980383290806892, 4.32039212045326519058636437187, 4.94284031943159442030500635338, 5.94629015825883792891429139086, 6.23186765922244566574282930469, 7.34689845150244954575088779748, 7.984189477416458754672061917520

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