Properties

Degree $2$
Conductor $328560$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s − 4·11-s + 2·13-s − 15-s − 2·17-s + 4·19-s − 8·23-s + 25-s + 27-s + 2·29-s + 8·31-s − 4·33-s + 2·39-s + 10·41-s + 12·43-s − 45-s − 7·49-s − 2·51-s + 6·53-s + 4·55-s + 4·57-s + 4·59-s + 10·61-s − 2·65-s + 4·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.258·15-s − 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s + 0.320·39-s + 1.56·41-s + 1.82·43-s − 0.149·45-s − 49-s − 0.280·51-s + 0.824·53-s + 0.539·55-s + 0.529·57-s + 0.520·59-s + 1.28·61-s − 0.248·65-s + 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(328560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 37^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{328560} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 328560,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
37 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.83729933908720, −12.32881898966781, −12.08862664811675, −11.27923341428237, −11.16829677681084, −10.51150159766073, −9.981237266075078, −9.788699176507728, −9.140732616595835, −8.543793080761388, −8.224742418776649, −7.836788730470677, −7.439816192455709, −6.881789034190549, −6.327047725977930, −5.673632183633607, −5.461450931503454, −4.589704928429461, −4.202521739286726, −3.862128801733935, −3.046855839378173, −2.632034010686063, −2.288381400154437, −1.376349213678937, −0.7980392206997696, 0, 0.7980392206997696, 1.376349213678937, 2.288381400154437, 2.632034010686063, 3.046855839378173, 3.862128801733935, 4.202521739286726, 4.589704928429461, 5.461450931503454, 5.673632183633607, 6.327047725977930, 6.881789034190549, 7.439816192455709, 7.836788730470677, 8.224742418776649, 8.543793080761388, 9.140732616595835, 9.788699176507728, 9.981237266075078, 10.51150159766073, 11.16829677681084, 11.27923341428237, 12.08862664811675, 12.32881898966781, 12.83729933908720

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