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 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 7-s − 8-s + 2·11-s + 2·13-s − 14-s + 16-s − 8·17-s − 2·19-s − 2·22-s − 2·26-s + 28-s + 6·29-s + 6·31-s − 32-s + 8·34-s + 8·37-s + 2·38-s − 6·41-s + 8·43-s + 2·44-s − 4·47-s + 49-s + 2·52-s + 2·53-s − 56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.603·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.94·17-s − 0.458·19-s − 0.426·22-s − 0.392·26-s + 0.188·28-s + 1.11·29-s + 1.07·31-s − 0.176·32-s + 1.37·34-s + 1.31·37-s + 0.324·38-s − 0.937·41-s + 1.21·43-s + 0.301·44-s − 0.583·47-s + 1/7·49-s + 0.277·52-s + 0.274·53-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  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.364997505$
$L(\frac12)$  $\approx$  $1.364997505$
$L(\frac{3}{2})$   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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
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

−8.531591349493028338806006163081, −8.295391565540689042343474124786, −7.12512042763986612212744454857, −6.56452419156117317297842184511, −5.92243413486258227392471516370, −4.66461903046678535954045988198, −4.09286529132425166137833371766, −2.80693794473626423997808224675, −1.94319432336838222034380704988, −0.794341975267530779418598313203, 0.794341975267530779418598313203, 1.94319432336838222034380704988, 2.80693794473626423997808224675, 4.09286529132425166137833371766, 4.66461903046678535954045988198, 5.92243413486258227392471516370, 6.56452419156117317297842184511, 7.12512042763986612212744454857, 8.295391565540689042343474124786, 8.531591349493028338806006163081

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