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 − 2·7-s + 9-s − 11-s − 13-s − 17-s − 3·19-s + 2·21-s − 4·23-s − 27-s − 7·31-s + 33-s − 37-s + 39-s + 4·41-s − 7·43-s − 10·47-s − 3·49-s + 51-s + 14·53-s + 3·57-s − 10·59-s + 5·61-s − 2·63-s + 4·69-s − 10·71-s + 16·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.242·17-s − 0.688·19-s + 0.436·21-s − 0.834·23-s − 0.192·27-s − 1.25·31-s + 0.174·33-s − 0.164·37-s + 0.160·39-s + 0.624·41-s − 1.06·43-s − 1.45·47-s − 3/7·49-s + 0.140·51-s + 1.92·53-s + 0.397·57-s − 1.30·59-s + 0.640·61-s − 0.251·63-s + 0.481·69-s − 1.18·71-s + 1.87·73-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 17 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
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

−13.08070367141265, −12.59016504440435, −11.98739142966490, −11.84807837438977, −11.10026219562208, −10.70182581818741, −10.36802034194396, −9.765867326230951, −9.459580951274313, −8.932108123972224, −8.312019666248900, −7.890436730664173, −7.316336613915604, −6.824772421434234, −6.346562765471707, −6.043318578526097, −5.336515254350951, −4.995591385592509, −4.379844986935678, −3.729874289032786, −3.432668386903021, −2.594903756889050, −2.086640319473239, −1.500592052951720, −0.5291711224656856, 0, 0.5291711224656856, 1.500592052951720, 2.086640319473239, 2.594903756889050, 3.432668386903021, 3.729874289032786, 4.379844986935678, 4.995591385592509, 5.336515254350951, 6.043318578526097, 6.346562765471707, 6.824772421434234, 7.316336613915604, 7.890436730664173, 8.312019666248900, 8.932108123972224, 9.459580951274313, 9.765867326230951, 10.36802034194396, 10.70182581818741, 11.10026219562208, 11.84807837438977, 11.98739142966490, 12.59016504440435, 13.08070367141265

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