Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 5 \cdot 13 \cdot 17 $
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 − 4·7-s + 9-s − 13-s − 15-s + 17-s + 4·19-s + 4·21-s + 25-s − 27-s − 6·29-s − 4·31-s − 4·35-s − 2·37-s + 39-s + 6·41-s + 4·43-s + 45-s + 9·49-s − 51-s − 6·53-s − 4·57-s + 12·59-s + 10·61-s − 4·63-s − 65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.258·15-s + 0.242·17-s + 0.917·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.676·35-s − 0.328·37-s + 0.160·39-s + 0.937·41-s + 0.609·43-s + 0.149·45-s + 9/7·49-s − 0.140·51-s − 0.824·53-s − 0.529·57-s + 1.56·59-s + 1.28·61-s − 0.503·63-s − 0.124·65-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(212160\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{212160} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 212160,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;13,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.11899654222508, −12.87763064929273, −12.33959734753916, −11.91336335541724, −11.37165296602086, −10.89772233568301, −10.38790598620448, −9.807868510272935, −9.657667934623950, −9.188727144408218, −8.667581738717855, −7.923778321235696, −7.257433330977398, −7.089231623285287, −6.496153277759458, −5.947597136930650, −5.467705143026101, −5.329962893438632, −4.296015075955476, −3.957626013345615, −3.225886115203201, −2.862688061064729, −2.137598743200888, −1.422543119852711, −0.6745566988871889, 0, 0.6745566988871889, 1.422543119852711, 2.137598743200888, 2.862688061064729, 3.225886115203201, 3.957626013345615, 4.296015075955476, 5.329962893438632, 5.467705143026101, 5.947597136930650, 6.496153277759458, 7.089231623285287, 7.257433330977398, 7.923778321235696, 8.667581738717855, 9.188727144408218, 9.657667934623950, 9.807868510272935, 10.38790598620448, 10.89772233568301, 11.37165296602086, 11.91336335541724, 12.33959734753916, 12.87763064929273, 13.11899654222508

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