Properties

Label 2-15600-1.1-c1-0-42
Degree $2$
Conductor $15600$
Sign $-1$
Analytic cond. $124.566$
Root an. cond. $11.1609$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 13-s + 6·17-s − 4·19-s − 27-s − 2·29-s + 2·37-s + 39-s − 2·41-s − 4·43-s + 4·47-s − 7·49-s − 6·51-s + 10·53-s + 4·57-s − 8·59-s − 2·61-s − 4·67-s + 12·71-s + 6·73-s + 81-s − 16·83-s + 2·87-s − 10·89-s − 2·97-s + 101-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.277·13-s + 1.45·17-s − 0.917·19-s − 0.192·27-s − 0.371·29-s + 0.328·37-s + 0.160·39-s − 0.312·41-s − 0.609·43-s + 0.583·47-s − 49-s − 0.840·51-s + 1.37·53-s + 0.529·57-s − 1.04·59-s − 0.256·61-s − 0.488·67-s + 1.42·71-s + 0.702·73-s + 1/9·81-s − 1.75·83-s + 0.214·87-s − 1.05·89-s − 0.203·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15600\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(124.566\)
Root analytic conductor: \(11.1609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{15600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 15600,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
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

−16.55112067952925, −15.70559286436714, −15.16128242790825, −14.70649706989103, −14.06123223735589, −13.50675357446466, −12.73115844063051, −12.41538009737150, −11.84682984521491, −11.18659852223554, −10.71424994849189, −9.944243231298543, −9.729123850567495, −8.799889825053151, −8.207311175167622, −7.557217814971721, −6.976376899528075, −6.269227770003869, −5.676904660142074, −5.107407616568471, −4.372130814615299, −3.667128551160968, −2.879061530206123, −1.935121691334052, −1.074982779785844, 0, 1.074982779785844, 1.935121691334052, 2.879061530206123, 3.667128551160968, 4.372130814615299, 5.107407616568471, 5.676904660142074, 6.269227770003869, 6.976376899528075, 7.557217814971721, 8.207311175167622, 8.799889825053151, 9.729123850567495, 9.944243231298543, 10.71424994849189, 11.18659852223554, 11.84682984521491, 12.41538009737150, 12.73115844063051, 13.50675357446466, 14.06123223735589, 14.70649706989103, 15.16128242790825, 15.70559286436714, 16.55112067952925

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