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 + 4·13-s + 14-s + 16-s + 6·17-s + 2·19-s − 4·26-s − 28-s + 6·29-s − 4·31-s − 32-s − 6·34-s − 2·37-s − 2·38-s − 6·41-s − 8·43-s − 12·47-s + 49-s + 4·52-s + 6·53-s + 56-s − 6·58-s + 6·59-s + 8·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.458·19-s − 0.784·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s − 0.328·37-s − 0.324·38-s − 0.937·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s + 0.554·52-s + 0.824·53-s + 0.133·56-s − 0.787·58-s + 0.781·59-s + 1.02·61-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  :  $\chi_{3150} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.368961465$
$L(\frac12)$  $\approx$  $1.368961465$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−18.31639189267445, −18.22221537614356, −17.27948105952995, −16.60108902943227, −16.17911425311074, −15.62974289042542, −14.80451224354978, −14.20179491326030, −13.40222037198806, −12.80677941946481, −11.89374728165456, −11.54559951547881, −10.57059710741026, −10.06051435984104, −9.482758210540421, −8.510181799390279, −8.190576101673579, −7.205686924122237, −6.566145073841420, −5.768605422136366, −4.999291175854982, −3.624046627132018, −3.165938386346177, −1.809533547918366, −0.8147828821219185, 0.8147828821219185, 1.809533547918366, 3.165938386346177, 3.624046627132018, 4.999291175854982, 5.768605422136366, 6.566145073841420, 7.205686924122237, 8.190576101673579, 8.510181799390279, 9.482758210540421, 10.06051435984104, 10.57059710741026, 11.54559951547881, 11.89374728165456, 12.80677941946481, 13.40222037198806, 14.20179491326030, 14.80451224354978, 15.62974289042542, 16.17911425311074, 16.60108902943227, 17.27948105952995, 18.22221537614356, 18.31639189267445

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