Properties

Label 2-172480-1.1-c1-0-87
Degree $2$
Conductor $172480$
Sign $1$
Analytic cond. $1377.25$
Root an. cond. $37.1114$
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  + 2·3-s + 5-s + 9-s − 11-s − 4·13-s + 2·15-s + 4·19-s + 6·23-s + 25-s − 4·27-s + 6·29-s + 8·31-s − 2·33-s − 2·37-s − 8·39-s − 6·41-s + 8·43-s + 45-s + 6·47-s + 6·53-s − 55-s + 8·57-s + 12·59-s + 2·61-s − 4·65-s − 10·67-s + 12·69-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.516·15-s + 0.917·19-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s + 1.43·31-s − 0.348·33-s − 0.328·37-s − 1.28·39-s − 0.937·41-s + 1.21·43-s + 0.149·45-s + 0.875·47-s + 0.824·53-s − 0.134·55-s + 1.05·57-s + 1.56·59-s + 0.256·61-s − 0.496·65-s − 1.22·67-s + 1.44·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 172480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(172480\)    =    \(2^{6} \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(1377.25\)
Root analytic conductor: \(37.1114\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 172480,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.004012187\)
\(L(\frac12)\) \(\approx\) \(5.004012187\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
11 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−13.47032158889108, −12.76934948475318, −12.29232261117880, −11.94847712706227, −11.28424501710068, −10.79450535202464, −10.09340824552229, −9.855305601614417, −9.440117354210022, −8.764178933257402, −8.570010469837603, −7.918859197029111, −7.522509579331648, −6.861678762157600, −6.680172241012388, −5.602840147425345, −5.426681826543264, −4.740082174750447, −4.245043738843101, −3.472449864506611, −2.927483369876837, −2.563501835500041, −2.149602165413296, −1.203071836314759, −0.6267267583731093, 0.6267267583731093, 1.203071836314759, 2.149602165413296, 2.563501835500041, 2.927483369876837, 3.472449864506611, 4.245043738843101, 4.740082174750447, 5.426681826543264, 5.602840147425345, 6.680172241012388, 6.861678762157600, 7.522509579331648, 7.918859197029111, 8.570010469837603, 8.764178933257402, 9.440117354210022, 9.855305601614417, 10.09340824552229, 10.79450535202464, 11.28424501710068, 11.94847712706227, 12.29232261117880, 12.76934948475318, 13.47032158889108

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