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 + 6.42·11-s − 13-s − 14-s + 16-s + 5.42·17-s + 8.42·19-s − 6.42·22-s + 3.42·23-s + 26-s + 28-s − 3.42·29-s − 1.42·31-s − 32-s − 5.42·34-s + 8.84·37-s − 8.42·38-s + 10.4·41-s + 7·43-s + 6.42·44-s − 3.42·46-s − 10.4·47-s + 49-s − 52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.377·7-s − 0.353·8-s + 1.93·11-s − 0.277·13-s − 0.267·14-s + 0.250·16-s + 1.31·17-s + 1.93·19-s − 1.36·22-s + 0.714·23-s + 0.196·26-s + 0.188·28-s − 0.635·29-s − 0.255·31-s − 0.176·32-s − 0.930·34-s + 1.45·37-s − 1.36·38-s + 1.62·41-s + 1.06·43-s + 0.968·44-s − 0.504·46-s − 1.52·47-s + 0.142·49-s − 0.138·52-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  =  $(2,\ 9450,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.302700770$
$L(\frac12)$  $\approx$  $2.302700770$
$L(\frac{3}{2})$   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 \( 1 - 6.42T + 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 - 5.42T + 17T^{2} \)
19 \( 1 - 8.42T + 19T^{2} \)
23 \( 1 - 3.42T + 23T^{2} \)
29 \( 1 + 3.42T + 29T^{2} \)
31 \( 1 + 1.42T + 31T^{2} \)
37 \( 1 - 8.84T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 - 7T + 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 + 5T + 53T^{2} \)
59 \( 1 - 7.42T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 - 3T + 67T^{2} \)
71 \( 1 - 5.42T + 71T^{2} \)
73 \( 1 + 2.42T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 7.57T + 83T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−7.55328977730697777566663347456, −7.30747627034380874196053803739, −6.42508496443011791599354783734, −5.74662113277053634053109974186, −5.07019661601460422570935549524, −4.03948931634693608489759938734, −3.42169317763331040942022624053, −2.53580149543667072576251740225, −1.28035722352257182438114929080, −1.01260825345890657038299204786, 1.01260825345890657038299204786, 1.28035722352257182438114929080, 2.53580149543667072576251740225, 3.42169317763331040942022624053, 4.03948931634693608489759938734, 5.07019661601460422570935549524, 5.74662113277053634053109974186, 6.42508496443011791599354783734, 7.30747627034380874196053803739, 7.55328977730697777566663347456

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