Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 5^{2} \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s + 9-s + 4·11-s − 2·13-s + 6·17-s + 4·19-s + 21-s − 27-s + 2·29-s − 4·33-s + 6·37-s + 2·39-s + 2·41-s + 4·43-s + 49-s − 6·51-s + 6·53-s − 4·57-s + 12·59-s + 2·61-s − 63-s − 4·67-s + 6·73-s − 4·77-s + 16·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.218·21-s − 0.192·27-s + 0.371·29-s − 0.696·33-s + 0.986·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s + 1/7·49-s − 0.840·51-s + 0.824·53-s − 0.529·57-s + 1.56·59-s + 0.256·61-s − 0.125·63-s − 0.488·67-s + 0.702·73-s − 0.455·77-s + 1.80·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(33600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{33600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 33600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.417232116$
$L(\frac12)$  $\approx$  $2.417232116$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−14.80430125866463, −14.65155755939188, −13.84387391288792, −13.59096735799922, −12.57307332757042, −12.40829950728620, −11.85216658348249, −11.40877075732401, −10.81154087425392, −10.02263817222484, −9.729496135738963, −9.297108511986945, −8.526768756806815, −7.821558088898063, −7.290231458818911, −6.771267127041178, −6.135923767757036, −5.586701123300880, −5.075183328190938, −4.251207184832909, −3.710244808182174, −3.041067267626762, −2.208653685269620, −1.160547307808244, −0.7247743354766993, 0.7247743354766993, 1.160547307808244, 2.208653685269620, 3.041067267626762, 3.710244808182174, 4.251207184832909, 5.075183328190938, 5.586701123300880, 6.135923767757036, 6.771267127041178, 7.290231458818911, 7.821558088898063, 8.526768756806815, 9.297108511986945, 9.729496135738963, 10.02263817222484, 10.81154087425392, 11.40877075732401, 11.85216658348249, 12.40829950728620, 12.57307332757042, 13.59096735799922, 13.84387391288792, 14.65155755939188, 14.80430125866463

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