Properties

Label 2-3920-1.1-c1-0-4
Degree $2$
Conductor $3920$
Sign $1$
Analytic cond. $31.3013$
Root an. cond. $5.59476$
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  − 3-s − 5-s − 2·9-s − 3·11-s − 13-s + 15-s + 5·17-s − 6·19-s + 25-s + 5·27-s − 5·29-s + 2·31-s + 3·33-s − 4·37-s + 39-s + 2·41-s − 10·43-s + 2·45-s − 9·47-s − 5·51-s + 6·53-s + 3·55-s + 6·57-s − 6·59-s + 12·61-s + 65-s + 2·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.904·11-s − 0.277·13-s + 0.258·15-s + 1.21·17-s − 1.37·19-s + 1/5·25-s + 0.962·27-s − 0.928·29-s + 0.359·31-s + 0.522·33-s − 0.657·37-s + 0.160·39-s + 0.312·41-s − 1.52·43-s + 0.298·45-s − 1.31·47-s − 0.700·51-s + 0.824·53-s + 0.404·55-s + 0.794·57-s − 0.781·59-s + 1.53·61-s + 0.124·65-s + 0.244·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(31.3013\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7286189015\)
\(L(\frac12)\) \(\approx\) \(0.7286189015\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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

−8.231230058258712000243092433127, −7.934819959822274081714681139985, −6.92288217725136468536000846238, −6.24242977831215165810735121848, −5.35318682894933643029769478154, −4.95702665735423568751724537401, −3.82239877858026214780958170553, −3.02250890666083408706942626208, −2.00393289721216725637791883713, −0.48300653155031577048533313748, 0.48300653155031577048533313748, 2.00393289721216725637791883713, 3.02250890666083408706942626208, 3.82239877858026214780958170553, 4.95702665735423568751724537401, 5.35318682894933643029769478154, 6.24242977831215165810735121848, 6.92288217725136468536000846238, 7.934819959822274081714681139985, 8.231230058258712000243092433127

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