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 − 13-s + 14-s + 16-s + 3·17-s − 7·19-s − 6·23-s − 26-s + 28-s + 3·29-s + 8·31-s + 32-s + 3·34-s + 2·37-s − 7·38-s + 6·41-s + 2·43-s − 6·46-s − 3·47-s + 49-s − 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 − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 1.60·19-s − 1.25·23-s − 0.196·26-s + 0.188·28-s + 0.557·29-s + 1.43·31-s + 0.176·32-s + 0.514·34-s + 0.328·37-s − 1.13·38-s + 0.937·41-s + 0.304·43-s − 0.884·46-s − 0.437·47-s + 1/7·49-s − 0.138·52-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  =  0
Selberg data  =  $(2,\ 9450,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.425937452$
$L(\frac12)$  $\approx$  $3.425937452$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
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 \( 1 + T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−7.897027663604322173678949232804, −6.80372035451057240722667487356, −6.27982096164805657207952288763, −5.67358607205667203911547653900, −4.79864064774061289985810246364, −4.29843115281266460124353530081, −3.58107677912591984673243535167, −2.56566243850023025022238641415, −2.00080981543475741545929929484, −0.789541050929285862978630047886, 0.789541050929285862978630047886, 2.00080981543475741545929929484, 2.56566243850023025022238641415, 3.58107677912591984673243535167, 4.29843115281266460124353530081, 4.79864064774061289985810246364, 5.67358607205667203911547653900, 6.27982096164805657207952288763, 6.80372035451057240722667487356, 7.897027663604322173678949232804

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