Properties

Label 2-388080-1.1-c1-0-130
Degree $2$
Conductor $388080$
Sign $-1$
Analytic cond. $3098.83$
Root an. cond. $55.6671$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 11-s − 6·13-s − 8·19-s − 6·23-s + 25-s − 4·29-s + 4·31-s + 4·37-s − 2·41-s + 12·43-s − 8·47-s − 4·53-s − 55-s − 10·59-s + 14·61-s + 6·65-s + 4·67-s + 8·71-s − 12·73-s + 2·79-s − 2·83-s + 12·89-s + 8·95-s + 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 1.66·13-s − 1.83·19-s − 1.25·23-s + 1/5·25-s − 0.742·29-s + 0.718·31-s + 0.657·37-s − 0.312·41-s + 1.82·43-s − 1.16·47-s − 0.549·53-s − 0.134·55-s − 1.30·59-s + 1.79·61-s + 0.744·65-s + 0.488·67-s + 0.949·71-s − 1.40·73-s + 0.225·79-s − 0.219·83-s + 1.27·89-s + 0.820·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(388080\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(3098.83\)
Root analytic conductor: \(55.6671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 388080,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 \)
11 \( 1 - T \)
good13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−12.77352099278330, −12.15308343316239, −11.89593220169757, −11.40424076422677, −10.75922501779710, −10.58801537798620, −9.881301150009505, −9.538697639039804, −9.218399421916689, −8.390101907220872, −8.173599110914095, −7.748587699369290, −7.119443312771364, −6.826421439234627, −6.134984101731414, −5.903226444798998, −5.084093433057956, −4.685287816215605, −4.166564175617269, −3.890073197804864, −3.109239036522639, −2.422910795052090, −2.194859102204392, −1.478853480994189, −0.5325053342473378, 0, 0.5325053342473378, 1.478853480994189, 2.194859102204392, 2.422910795052090, 3.109239036522639, 3.890073197804864, 4.166564175617269, 4.685287816215605, 5.084093433057956, 5.903226444798998, 6.134984101731414, 6.826421439234627, 7.119443312771364, 7.748587699369290, 8.173599110914095, 8.390101907220872, 9.218399421916689, 9.538697639039804, 9.881301150009505, 10.58801537798620, 10.75922501779710, 11.40424076422677, 11.89593220169757, 12.15308343316239, 12.77352099278330

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