Properties

Degree $2$
Conductor $3150$
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 + 2·13-s − 14-s + 16-s + 2·17-s − 4·19-s − 4·22-s − 8·23-s + 2·26-s − 28-s − 6·29-s − 8·31-s + 32-s + 2·34-s + 2·37-s − 4·38-s − 2·41-s + 12·43-s − 4·44-s − 8·46-s − 8·47-s + 49-s + 2·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.20·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.852·22-s − 1.66·23-s + 0.392·26-s − 0.188·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.342·34-s + 0.328·37-s − 0.648·38-s − 0.312·41-s + 1.82·43-s − 0.603·44-s − 1.17·46-s − 1.16·47-s + 1/7·49-s + 0.277·52-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

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{3150} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3150,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.80204045797964, −18.29857755560516, −17.69422765827392, −16.69156014535305, −16.24775831053223, −15.77030190285734, −14.95397234989436, −14.54041652701749, −13.55835299295362, −13.27005527112967, −12.50808089382131, −12.05873778352821, −10.94834482570328, −10.71642769643197, −9.842894865668311, −9.078646662439493, −8.038242046145280, −7.641489860414607, −6.654846723237199, −5.828451484487465, −5.429109344832441, −4.273492778490803, −3.662247559812708, −2.661363636053908, −1.782345356954266, 0, 1.782345356954266, 2.661363636053908, 3.662247559812708, 4.273492778490803, 5.429109344832441, 5.828451484487465, 6.654846723237199, 7.641489860414607, 8.038242046145280, 9.078646662439493, 9.842894865668311, 10.71642769643197, 10.94834482570328, 12.05873778352821, 12.50808089382131, 13.27005527112967, 13.55835299295362, 14.54041652701749, 14.95397234989436, 15.77030190285734, 16.24775831053223, 16.69156014535305, 17.69422765827392, 18.29857755560516, 18.80204045797964

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