↑

Properties

Label 2-331056-1.1-c1-0-110
Degree $2$
Conductor $331056$
Sign $1$
Analytic cond. $2643.49$
Root an. cond. $51.4149$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 4·7-s + 4·13-s − 3·17-s + 19-s − 6·23-s + 4·25-s − 9·29-s − 8·31-s + 12·35-s − 7·37-s − 6·41-s − 43-s + 6·47-s + 9·49-s − 6·59-s + 61-s − 12·65-s + 7·67-s + 12·71-s + 10·73-s + 8·79-s + 6·83-s + 9·85-s − 15·89-s − 16·91-s − 3·95-s + ⋯
L(s)  = 1  − 1.34·5-s − 1.51·7-s + 1.10·13-s − 0.727·17-s + 0.229·19-s − 1.25·23-s + 4/5·25-s − 1.67·29-s − 1.43·31-s + 2.02·35-s − 1.15·37-s − 0.937·41-s − 0.152·43-s + 0.875·47-s + 9/7·49-s − 0.781·59-s + 0.128·61-s − 1.48·65-s + 0.855·67-s + 1.42·71-s + 1.17·73-s + 0.900·79-s + 0.658·83-s + 0.976·85-s − 1.58·89-s − 1.67·91-s − 0.307·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(331056\)    =    \(2^{4} \cdot 3^{2} \cdot 11^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(2643.49\)
Root analytic conductor: \(51.4149\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 331056,\ (\ :1/2),\ 1)\)

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 \( 1 \)
11 \( 1 \)
19 \( 1 - T \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
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

−12.96207361885477, −12.52899108190412, −12.24576546732890, −11.70373647084980, −11.14851506587599, −10.93549351315027, −10.42402825226630, −9.758529747993214, −9.405164918958995, −8.938371704628878, −8.462296690670045, −7.998700933926604, −7.507239757920374, −6.915051820732961, −6.750798945163598, −6.014798096349462, −5.691677678813534, −5.065578435485163, −4.290341953423654, −3.788857074663380, −3.493985581838492, −3.341209781004101, −2.259156399304788, −1.869965906986307, −0.8843514763732783, 0, 0, 0.8843514763732783, 1.869965906986307, 2.259156399304788, 3.341209781004101, 3.493985581838492, 3.788857074663380, 4.290341953423654, 5.065578435485163, 5.691677678813534, 6.014798096349462, 6.750798945163598, 6.915051820732961, 7.507239757920374, 7.998700933926604, 8.462296690670045, 8.938371704628878, 9.405164918958995, 9.758529747993214, 10.42402825226630, 10.93549351315027, 11.14851506587599, 11.70373647084980, 12.24576546732890, 12.52899108190412, 12.96207361885477

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