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 − 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  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.124854673$
$L(\frac12)$  $\approx$  $3.124854673$
$L(\frac{3}{2})$   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

−8.559445614310455916494758321079, −7.80516974580249205703440909183, −7.14253006005762024799153823515, −6.39860842539494741491816824731, −5.35395270183027751713079831508, −4.99694463086816364799307883145, −4.04206921003813732100393271148, −3.05783796505154958047042891710, −2.30836202040639030126924935645, −0.994854636871361997332992172547, 0.994854636871361997332992172547, 2.30836202040639030126924935645, 3.05783796505154958047042891710, 4.04206921003813732100393271148, 4.99694463086816364799307883145, 5.35395270183027751713079831508, 6.39860842539494741491816824731, 7.14253006005762024799153823515, 7.80516974580249205703440909183, 8.559445614310455916494758321079

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