Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \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 + 5·11-s − 6·13-s + 14-s + 16-s + 17-s − 3·19-s − 5·22-s + 6·26-s − 28-s + 6·29-s − 4·31-s − 32-s − 34-s + 8·37-s + 3·38-s − 11·41-s − 8·43-s + 5·44-s − 2·47-s + 49-s − 6·52-s − 4·53-s + 56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.50·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.688·19-s − 1.06·22-s + 1.17·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.171·34-s + 1.31·37-s + 0.486·38-s − 1.71·41-s − 1.21·43-s + 0.753·44-s − 0.291·47-s + 1/7·49-s − 0.832·52-s − 0.549·53-s + 0.133·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−8.387182329094963452351149311648, −7.60593169459124745854077628187, −6.71238157904374327273470767010, −6.46479582570974087018240392764, −5.26015307963936998972992039268, −4.39148677091846193085504832465, −3.40111091922544740626092357501, −2.42817721651914966878410253933, −1.39287136921494160793057743506, 0, 1.39287136921494160793057743506, 2.42817721651914966878410253933, 3.40111091922544740626092357501, 4.39148677091846193085504832465, 5.26015307963936998972992039268, 6.46479582570974087018240392764, 6.71238157904374327273470767010, 7.60593169459124745854077628187, 8.387182329094963452351149311648

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