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

−13.38140764507325, −12.74795447759782, −12.38283911817519, −11.84841958210014, −11.27150058270643, −10.91263256201324, −10.51914485540680, −9.880743727304298, −9.477349407323663, −8.942846020705090, −8.494909281957523, −7.938216268453678, −7.797470207436480, −7.079175049100427, −6.598151798341134, −5.968351037498245, −5.395857747897064, −4.966202016804956, −4.394015602266879, −4.021428932682532, −3.261751303036941, −2.607685400304515, −2.076797582869755, −1.622065917284668, −1.053509115452291, 0, 1.053509115452291, 1.622065917284668, 2.076797582869755, 2.607685400304515, 3.261751303036941, 4.021428932682532, 4.394015602266879, 4.966202016804956, 5.395857747897064, 5.968351037498245, 6.598151798341134, 7.079175049100427, 7.797470207436480, 7.938216268453678, 8.494909281957523, 8.942846020705090, 9.477349407323663, 9.880743727304298, 10.51914485540680, 10.91263256201324, 11.27150058270643, 11.84841958210014, 12.38283911817519, 12.74795447759782, 13.38140764507325

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