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 + 4·11-s − 3·13-s − 14-s + 16-s − 7·17-s − 6·19-s + 4·22-s − 9·23-s − 3·26-s − 28-s + 3·29-s − 7·31-s + 32-s − 7·34-s − 10·37-s − 6·38-s − 41-s + 13·43-s + 4·44-s − 9·46-s + 2·47-s + 49-s − 3·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 1.20·11-s − 0.832·13-s − 0.267·14-s + 1/4·16-s − 1.69·17-s − 1.37·19-s + 0.852·22-s − 1.87·23-s − 0.588·26-s − 0.188·28-s + 0.557·29-s − 1.25·31-s + 0.176·32-s − 1.20·34-s − 1.64·37-s − 0.973·38-s − 0.156·41-s + 1.98·43-s + 0.603·44-s − 1.32·46-s + 0.291·47-s + 1/7·49-s − 0.416·52-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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - 13 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−8.364570454178253922480972676055, −7.29939558493450902632297012573, −6.62761586986746228044789366008, −6.17551464616597275862107485654, −5.18582660634471050001288987818, −4.08966430069560043880481135460, −3.96984208348937638316720060401, −2.50083405929196491803802214169, −1.86043516958039461064905083861, 0, 1.86043516958039461064905083861, 2.50083405929196491803802214169, 3.96984208348937638316720060401, 4.08966430069560043880481135460, 5.18582660634471050001288987818, 6.17551464616597275862107485654, 6.62761586986746228044789366008, 7.29939558493450902632297012573, 8.364570454178253922480972676055

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