Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{2} \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 − 4·7-s + 9-s + 4·11-s − 13-s + 17-s + 4·19-s + 4·21-s − 27-s − 6·29-s − 4·33-s + 10·37-s + 39-s − 10·41-s − 12·43-s − 8·47-s + 9·49-s − 51-s − 2·53-s − 4·57-s − 2·61-s − 4·63-s + 4·67-s + 8·71-s − 2·73-s − 16·77-s + 4·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.242·17-s + 0.917·19-s + 0.872·21-s − 0.192·27-s − 1.11·29-s − 0.696·33-s + 1.64·37-s + 0.160·39-s − 1.56·41-s − 1.82·43-s − 1.16·47-s + 9/7·49-s − 0.140·51-s − 0.274·53-s − 0.529·57-s − 0.256·61-s − 0.503·63-s + 0.488·67-s + 0.949·71-s − 0.234·73-s − 1.82·77-s + 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(265200\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{265200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 265200,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 \)
13 \( 1 + T \)
17 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.07682640618770, −12.51806492520855, −12.01754664364054, −11.75598495078221, −11.29753319006956, −10.76584682531379, −10.10445741548785, −9.699345453529989, −9.506185376331044, −9.113233761878561, −8.303527282558605, −7.896183617097921, −7.183566970203446, −6.780323811294551, −6.449098423988939, −6.061936417718565, −5.352716651963259, −5.054419435234414, −4.259762998827662, −3.741673384823087, −3.313862317899556, −2.887571262267734, −1.950943774881051, −1.408468040853977, −0.6603122037485992, 0, 0.6603122037485992, 1.408468040853977, 1.950943774881051, 2.887571262267734, 3.313862317899556, 3.741673384823087, 4.259762998827662, 5.054419435234414, 5.352716651963259, 6.061936417718565, 6.449098423988939, 6.780323811294551, 7.183566970203446, 7.896183617097921, 8.303527282558605, 9.113233761878561, 9.506185376331044, 9.699345453529989, 10.10445741548785, 10.76584682531379, 11.29753319006956, 11.75598495078221, 12.01754664364054, 12.51806492520855, 13.07682640618770

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