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 + 5·11-s + 6·13-s + 14-s + 16-s − 17-s − 3·19-s + 5·22-s + 6·26-s + 28-s + 6·29-s − 4·31-s + 32-s − 34-s − 8·37-s − 3·38-s − 11·41-s + 8·43-s + 5·44-s + 2·47-s + 49-s + 6·52-s + 4·53-s + 56-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.50·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.688·19-s + 1.06·22-s + 1.17·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.171·34-s − 1.31·37-s − 0.486·38-s − 1.71·41-s + 1.21·43-s + 0.753·44-s + 0.291·47-s + 1/7·49-s + 0.832·52-s + 0.549·53-s + 0.133·56-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  =  $(2,\ 3150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.683106985$
$L(\frac12)$  $\approx$  $3.683106985$
$L(\frac{3}{2})$   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 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
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

−18.63711833402818, −17.81460845240008, −17.24958290645752, −16.60485001677886, −15.94218738925375, −15.30990340576472, −14.73652440524709, −13.87028481417233, −13.77369830802164, −12.82002378654424, −12.08154409757399, −11.62689227513251, −10.82681625280628, −10.42358016971919, −9.153007626061091, −8.745835108946328, −7.967535843941389, −6.828337128531473, −6.461770382306047, −5.687281630640304, −4.714858126009276, −3.917361854031801, −3.399306302846803, −2.012836347892483, −1.164229935900348, 1.164229935900348, 2.012836347892483, 3.399306302846803, 3.917361854031801, 4.714858126009276, 5.687281630640304, 6.461770382306047, 6.828337128531473, 7.967535843941389, 8.745835108946328, 9.153007626061091, 10.42358016971919, 10.82681625280628, 11.62689227513251, 12.08154409757399, 12.82002378654424, 13.77369830802164, 13.87028481417233, 14.73652440524709, 15.30990340576472, 15.94218738925375, 16.60485001677886, 17.24958290645752, 17.81460845240008, 18.63711833402818

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