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 − 8·23-s − 27-s − 2·29-s + 4·31-s − 4·33-s + 6·37-s + 39-s − 6·41-s − 4·43-s − 4·47-s + 9·49-s + 51-s − 14·53-s − 4·57-s − 10·61-s − 4·63-s − 12·67-s + 8·69-s + 8·71-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 − 1.66·23-s − 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.696·33-s + 0.986·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s − 0.583·47-s + 9/7·49-s + 0.140·51-s − 1.92·53-s − 0.529·57-s − 1.28·61-s − 0.503·63-s − 1.46·67-s + 0.963·69-s + 0.949·71-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 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
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

−13.02421241070121, −12.41834593760080, −12.12921119622932, −11.71922894716115, −11.33740717332754, −10.70980423219161, −10.09523479037670, −9.764289138255915, −9.519049475543684, −9.039661196809931, −8.370837945662260, −7.790538267336867, −7.347633683551325, −6.621232434560534, −6.476457262400938, −6.023747131462390, −5.591216815790737, −4.757261678587430, −4.415286163759676, −3.732748876777192, −3.320865515777081, −2.849968670636389, −1.948860171596337, −1.456496145422203, −0.6085165902251736, 0, 0.6085165902251736, 1.456496145422203, 1.948860171596337, 2.849968670636389, 3.320865515777081, 3.732748876777192, 4.415286163759676, 4.757261678587430, 5.591216815790737, 6.023747131462390, 6.476457262400938, 6.621232434560534, 7.347633683551325, 7.790538267336867, 8.370837945662260, 9.039661196809931, 9.519049475543684, 9.764289138255915, 10.09523479037670, 10.70980423219161, 11.33740717332754, 11.71922894716115, 12.12921119622932, 12.41834593760080, 13.02421241070121

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