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 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 7-s + 8-s − 7·13-s + 14-s + 16-s − 3·17-s − 19-s + 6·23-s − 7·26-s + 28-s + 3·29-s − 4·31-s + 32-s − 3·34-s + 2·37-s − 38-s − 6·41-s + 2·43-s + 6·46-s + 3·47-s + 49-s − 7·52-s + 9·53-s + 56-s + 3·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 1.94·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.229·19-s + 1.25·23-s − 1.37·26-s + 0.188·28-s + 0.557·29-s − 0.718·31-s + 0.176·32-s − 0.514·34-s + 0.328·37-s − 0.162·38-s − 0.937·41-s + 0.304·43-s + 0.884·46-s + 0.437·47-s + 1/7·49-s − 0.970·52-s + 1.23·53-s + 0.133·56-s + 0.393·58-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  =  1
Selberg data  =  $(2,\ 9450,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−7.21372180010215654252745918078, −6.79412967415503416242406801210, −5.86541918866092582633004991010, −5.10274185521135129723767956032, −4.69508036462649347825894031805, −4.00209041095410698766008005561, −2.88734375994309713503561276188, −2.44138305098421678240160356990, −1.43566032589665524067972287384, 0, 1.43566032589665524067972287384, 2.44138305098421678240160356990, 2.88734375994309713503561276188, 4.00209041095410698766008005561, 4.69508036462649347825894031805, 5.10274185521135129723767956032, 5.86541918866092582633004991010, 6.79412967415503416242406801210, 7.21372180010215654252745918078

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