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  =  $(2,\ 369600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.946174495$
$L(\frac12)$  $\approx$  $2.946174495$
$L(\frac{3}{2})$   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

−12.41049736754567, −11.97177406454780, −11.70275593926707, −10.99946033319652, −10.70810344933662, −10.51395524340406, −9.758550611458692, −9.328175261773529, −8.894972054587583, −8.380920782462092, −8.033705702505182, −7.406762926079374, −6.717550696426142, −6.485118686159221, −6.002813849071323, −5.751030430545963, −4.958961797951358, −4.364013844898827, −4.086793517504006, −3.569100084911917, −2.833340616089393, −2.344553467792423, −1.627745177877646, −0.8870998946472962, −0.5964649351233395, 0.5964649351233395, 0.8870998946472962, 1.627745177877646, 2.344553467792423, 2.833340616089393, 3.569100084911917, 4.086793517504006, 4.364013844898827, 4.958961797951358, 5.751030430545963, 6.002813849071323, 6.485118686159221, 6.717550696426142, 7.406762926079374, 8.033705702505182, 8.380920782462092, 8.894972054587583, 9.328175261773529, 9.758550611458692, 10.51395524340406, 10.70810344933662, 10.99946033319652, 11.70275593926707, 11.97177406454780, 12.41049736754567

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