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 − 2·11-s + 13-s + 14-s + 16-s + 17-s + 4·19-s + 2·22-s − 7·23-s − 26-s − 28-s − 29-s + 3·31-s − 32-s − 34-s − 6·37-s − 4·38-s + 3·41-s − 43-s − 2·44-s + 7·46-s + 12·47-s + 49-s + 52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.603·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.917·19-s + 0.426·22-s − 1.45·23-s − 0.196·26-s − 0.188·28-s − 0.185·29-s + 0.538·31-s − 0.176·32-s − 0.171·34-s − 0.986·37-s − 0.648·38-s + 0.468·41-s − 0.152·43-s − 0.301·44-s + 1.03·46-s + 1.75·47-s + 1/7·49-s + 0.138·52-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−8.296532418800698668032493796347, −7.64817800946497920285515061772, −7.00292082281195566118758927168, −6.02886062140728316323118853364, −5.50662563248420743518416375774, −4.33428105829599150659696726487, −3.35139702087723593517618361152, −2.49900580864402951337675980753, −1.35741366069724757322673710974, 0, 1.35741366069724757322673710974, 2.49900580864402951337675980753, 3.35139702087723593517618361152, 4.33428105829599150659696726487, 5.50662563248420743518416375774, 6.02886062140728316323118853364, 7.00292082281195566118758927168, 7.64817800946497920285515061772, 8.296532418800698668032493796347

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