Properties

Label 2-9600-1.1-c1-0-123
Degree $2$
Conductor $9600$
Sign $-1$
Analytic cond. $76.6563$
Root an. cond. $8.75536$
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 + 4·7-s + 9-s − 6·13-s + 2·17-s + 6·19-s − 4·21-s + 6·23-s − 27-s − 8·29-s − 8·31-s − 10·37-s + 6·39-s − 6·41-s − 4·43-s + 2·47-s + 9·49-s − 2·51-s + 6·53-s − 6·57-s − 12·59-s + 14·61-s + 4·63-s − 4·67-s − 6·69-s − 8·71-s + 4·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.66·13-s + 0.485·17-s + 1.37·19-s − 0.872·21-s + 1.25·23-s − 0.192·27-s − 1.48·29-s − 1.43·31-s − 1.64·37-s + 0.960·39-s − 0.937·41-s − 0.609·43-s + 0.291·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s − 0.794·57-s − 1.56·59-s + 1.79·61-s + 0.503·63-s − 0.488·67-s − 0.722·69-s − 0.949·71-s + 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9600\)    =    \(2^{7} \cdot 3 \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(76.6563\)
Root analytic conductor: \(8.75536\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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

−7.27164703786466815254244515491, −7.01577500663188624461067364029, −5.60347160058462239857795768551, −5.22155861062169079942738349089, −4.92582063888445896708531020011, −3.91710000355542786784808696998, −3.02910352938422244199176075785, −1.94535736456767261870444557183, −1.31955818041062224410620205756, 0, 1.31955818041062224410620205756, 1.94535736456767261870444557183, 3.02910352938422244199176075785, 3.91710000355542786784808696998, 4.92582063888445896708531020011, 5.22155861062169079942738349089, 5.60347160058462239857795768551, 7.01577500663188624461067364029, 7.27164703786466815254244515491

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