Properties

Label 2-334620-1.1-c1-0-31
Degree $2$
Conductor $334620$
Sign $1$
Analytic cond. $2671.95$
Root an. cond. $51.6909$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 4·7-s + 11-s + 6·17-s − 4·19-s + 4·23-s + 25-s − 2·29-s + 4·31-s + 4·35-s − 2·37-s + 8·41-s − 12·43-s + 9·49-s + 55-s + 4·59-s + 6·61-s + 6·67-s − 2·73-s + 4·77-s − 10·79-s + 6·83-s + 6·85-s + 6·89-s − 4·95-s + 6·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s + 0.301·11-s + 1.45·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s + 0.676·35-s − 0.328·37-s + 1.24·41-s − 1.82·43-s + 9/7·49-s + 0.134·55-s + 0.520·59-s + 0.768·61-s + 0.733·67-s − 0.234·73-s + 0.455·77-s − 1.12·79-s + 0.658·83-s + 0.650·85-s + 0.635·89-s − 0.410·95-s + 0.609·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(334620\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(2671.95\)
Root analytic conductor: \(51.6909\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 334620,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.031943597\)
\(L(\frac12)\) \(\approx\) \(5.031943597\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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

−12.56558637304338, −12.03068702640340, −11.70967290992974, −11.20339836812560, −10.83504602369724, −10.33388073834900, −9.893590137622079, −9.455334042811070, −8.776591022810544, −8.484411422563388, −8.018535316893929, −7.604687601631135, −7.029781682232605, −6.573978538879062, −5.963590713122116, −5.444968937037789, −5.100437751099668, −4.594529495660213, −4.087775624804838, −3.460579248246720, −2.883851914566656, −2.193195334663458, −1.720252740192683, −1.184496812890090, −0.6228083069907842, 0.6228083069907842, 1.184496812890090, 1.720252740192683, 2.193195334663458, 2.883851914566656, 3.460579248246720, 4.087775624804838, 4.594529495660213, 5.100437751099668, 5.444968937037789, 5.963590713122116, 6.573978538879062, 7.029781682232605, 7.604687601631135, 8.018535316893929, 8.484411422563388, 8.776591022810544, 9.455334042811070, 9.893590137622079, 10.33388073834900, 10.83504602369724, 11.20339836812560, 11.70967290992974, 12.03068702640340, 12.56558637304338

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