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 + 7·13-s − 14-s + 16-s + 7·17-s + 8·19-s − 2·22-s + 5·23-s − 7·26-s + 28-s − 9·29-s + 31-s − 32-s − 7·34-s − 2·37-s − 8·38-s − 11·41-s + 3·43-s + 2·44-s − 5·46-s − 4·47-s + 49-s + 7·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.603·11-s + 1.94·13-s − 0.267·14-s + 1/4·16-s + 1.69·17-s + 1.83·19-s − 0.426·22-s + 1.04·23-s − 1.37·26-s + 0.188·28-s − 1.67·29-s + 0.179·31-s − 0.176·32-s − 1.20·34-s − 0.328·37-s − 1.29·38-s − 1.71·41-s + 0.457·43-s + 0.301·44-s − 0.737·46-s − 0.583·47-s + 1/7·49-s + 0.970·52-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  :  $\chi_{3150} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.853888543$
$L(\frac12)$  $\approx$  $1.853888543$
$L(\frac{3}{2})$   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 \( 1 - 7 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
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

−18.55587319785769, −18.12100501849088, −17.24634712610749, −16.67168197773551, −16.23945518435384, −15.47604477573423, −14.89878247722719, −14.05586839338111, −13.61244217053790, −12.74274656283585, −11.82710387902781, −11.47298546490856, −10.78379459790400, −10.04213065368498, −9.247846868606505, −8.821648821316085, −7.860874397242278, −7.469036305378010, −6.472011547859011, −5.734438842141789, −5.035329577601047, −3.523994533725670, −3.323542510104303, −1.572593100288880, −1.067850291575022, 1.067850291575022, 1.572593100288880, 3.323542510104303, 3.523994533725670, 5.035329577601047, 5.734438842141789, 6.472011547859011, 7.469036305378010, 7.860874397242278, 8.821648821316085, 9.247846868606505, 10.04213065368498, 10.78379459790400, 11.47298546490856, 11.82710387902781, 12.74274656283585, 13.61244217053790, 14.05586839338111, 14.89878247722719, 15.47604477573423, 16.23945518435384, 16.67168197773551, 17.24634712610749, 18.12100501849088, 18.55587319785769

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