Properties

Degree 2
Conductor $ 2 \cdot 3^{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  + 2-s + 4-s + 7-s + 8-s + 11-s + 5·13-s + 14-s + 16-s − 4·17-s + 7·19-s + 22-s + 5·26-s + 28-s − 6·29-s + 32-s − 4·34-s − 4·37-s + 7·38-s − 3·41-s + 11·43-s + 44-s + 47-s + 49-s + 5·52-s + 53-s + 56-s − 6·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.301·11-s + 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s + 1.60·19-s + 0.213·22-s + 0.980·26-s + 0.188·28-s − 1.11·29-s + 0.176·32-s − 0.685·34-s − 0.657·37-s + 1.13·38-s − 0.468·41-s + 1.67·43-s + 0.150·44-s + 0.145·47-s + 1/7·49-s + 0.693·52-s + 0.137·53-s + 0.133·56-s − 0.787·58-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9450\)    =    \(2 \cdot 3^{3} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{9450} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 9450,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.132585870$
$L(\frac12)$  $\approx$  $4.132585870$
$L(\frac{3}{2})$   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 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
good11 \( 1 - T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
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

−7.69281721002622315616091221974, −6.80845046589559813447482285302, −6.37864901793928866415289483412, −5.42686166362065127311771623046, −5.14390592081714508072053021669, −3.91607667688486460192432538112, −3.77786565625704649931489889900, −2.70112667558685491883558397688, −1.79599909564359746541011864137, −0.920612456787474847026333699856, 0.920612456787474847026333699856, 1.79599909564359746541011864137, 2.70112667558685491883558397688, 3.77786565625704649931489889900, 3.91607667688486460192432538112, 5.14390592081714508072053021669, 5.42686166362065127311771623046, 6.37864901793928866415289483412, 6.80845046589559813447482285302, 7.69281721002622315616091221974

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