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 − 2·13-s + 14-s + 16-s − 6·17-s + 8·19-s + 2·26-s − 28-s − 6·29-s − 4·31-s − 32-s + 6·34-s + 10·37-s − 8·38-s + 6·41-s + 4·43-s + 49-s − 2·52-s − 6·53-s + 56-s + 6·58-s + 12·59-s − 10·61-s + 4·62-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 1.83·19-s + 0.392·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 1.64·37-s − 1.29·38-s + 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.277·52-s − 0.824·53-s + 0.133·56-s + 0.787·58-s + 1.56·59-s − 1.28·61-s + 0.508·62-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  =  \((2,\ 3150,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.044053307\)
\(L(\frac12)\)  \(\approx\)  \(1.044053307\)
\(L(\frac{3}{2})\)   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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−18.34940405918671, −18.01040190493068, −17.42105037335897, −16.58838852497814, −16.18645798431675, −15.54260808958262, −14.88761642289850, −14.19482440086888, −13.35127807016762, −12.83171547884773, −12.00286772813106, −11.29428162384077, −10.86942246934817, −9.859124186946253, −9.370859383721325, −8.923782644459482, −7.731433224544062, −7.449242140386271, −6.543300617659325, −5.789700244043097, −4.897800538258453, −3.867928200229114, −2.875179275295274, −2.010947066718422, −0.6705228448831212, 0.6705228448831212, 2.010947066718422, 2.875179275295274, 3.867928200229114, 4.897800538258453, 5.789700244043097, 6.543300617659325, 7.449242140386271, 7.731433224544062, 8.923782644459482, 9.370859383721325, 9.859124186946253, 10.86942246934817, 11.29428162384077, 12.00286772813106, 12.83171547884773, 13.35127807016762, 14.19482440086888, 14.88761642289850, 15.54260808958262, 16.18645798431675, 16.58838852497814, 17.42105037335897, 18.01040190493068, 18.34940405918671

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