Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 $
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 − 11-s + 6·13-s − 2·17-s + 4·19-s + 21-s + 27-s + 2·29-s − 8·31-s − 33-s + 6·37-s + 6·39-s + 10·41-s + 4·43-s − 8·47-s + 49-s − 2·51-s + 6·53-s + 4·57-s + 4·59-s + 10·61-s + 63-s + 12·67-s − 2·73-s − 77-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s − 0.485·17-s + 0.917·19-s + 0.218·21-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.174·33-s + 0.986·37-s + 0.960·39-s + 1.56·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s + 0.529·57-s + 0.520·59-s + 1.28·61-s + 0.125·63-s + 1.46·67-s − 0.234·73-s − 0.113·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{369600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 369600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.001070569$
$L(\frac12)$  $\approx$  $5.001070569$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.78933267531251, −11.94665801464131, −11.47406925291648, −11.18223739603059, −10.77894797579460, −10.25151861933846, −9.711366923694683, −9.255094810835934, −8.811654734301419, −8.481896915960386, −7.850702481602405, −7.670277355591038, −6.966257855480443, −6.583201387290973, −5.892379420424605, −5.582322822214837, −5.018705135788658, −4.343396931691885, −3.881152474295599, −3.547391870837481, −2.821668108564623, −2.383425057970615, −1.711612182100796, −1.128431056052207, −0.6026389059879007, 0.6026389059879007, 1.128431056052207, 1.711612182100796, 2.383425057970615, 2.821668108564623, 3.547391870837481, 3.881152474295599, 4.343396931691885, 5.018705135788658, 5.582322822214837, 5.892379420424605, 6.583201387290973, 6.966257855480443, 7.670277355591038, 7.850702481602405, 8.481896915960386, 8.811654734301419, 9.255094810835934, 9.711366923694683, 10.25151861933846, 10.77894797579460, 11.18223739603059, 11.47406925291648, 11.94665801464131, 12.78933267531251

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