Properties

Label 2-3120-1.1-c1-0-13
Degree $2$
Conductor $3120$
Sign $1$
Analytic cond. $24.9133$
Root an. cond. $4.99132$
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 − 4·7-s + 9-s − 4·11-s + 13-s + 15-s + 6·17-s − 4·21-s + 4·23-s + 25-s + 27-s − 6·29-s + 8·31-s − 4·33-s − 4·35-s − 2·37-s + 39-s + 10·41-s + 4·43-s + 45-s − 8·47-s + 9·49-s + 6·51-s − 2·53-s − 4·55-s − 4·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 0.258·15-s + 1.45·17-s − 0.872·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.696·33-s − 0.676·35-s − 0.328·37-s + 0.160·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s − 1.16·47-s + 9/7·49-s + 0.840·51-s − 0.274·53-s − 0.539·55-s − 0.520·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(24.9133\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3120} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.999513610\)
\(L(\frac12)\) \(\approx\) \(1.999513610\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−8.754517467301115594054212589429, −7.904196823378737509514841477457, −7.28973502428521264317570031619, −6.39325775668224731478952879757, −5.73420885996127355362762671519, −4.92470869103303092921596170306, −3.65539023724033520959040943289, −3.07333847405991902784353497722, −2.32376617368210121865545121077, −0.821890830630328138357339916930, 0.821890830630328138357339916930, 2.32376617368210121865545121077, 3.07333847405991902784353497722, 3.65539023724033520959040943289, 4.92470869103303092921596170306, 5.73420885996127355362762671519, 6.39325775668224731478952879757, 7.28973502428521264317570031619, 7.904196823378737509514841477457, 8.754517467301115594054212589429

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