Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \cdot 7 $
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 − 5-s + 7-s + 3·8-s + 10-s − 6·13-s − 14-s − 16-s − 2·17-s − 8·19-s + 20-s − 8·23-s + 25-s + 6·26-s − 28-s + 2·29-s + 4·31-s − 5·32-s + 2·34-s − 35-s − 2·37-s + 8·38-s − 3·40-s + 6·41-s + 4·43-s + 8·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.377·7-s + 1.06·8-s + 0.316·10-s − 1.66·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s − 1.83·19-s + 0.223·20-s − 1.66·23-s + 1/5·25-s + 1.17·26-s − 0.188·28-s + 0.371·29-s + 0.718·31-s − 0.883·32-s + 0.342·34-s − 0.169·35-s − 0.328·37-s + 1.29·38-s − 0.474·40-s + 0.937·41-s + 0.609·43-s + 1.17·46-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{315} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 315,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−19.91606783402330, −19.51281592064860, −18.84623626477553, −17.79385769344014, −17.37589593635053, −16.60141841325293, −15.59156807507789, −14.63125822495919, −14.01653423556724, −12.81955495459951, −12.14616784601024, −10.96971784880108, −10.15928312968821, −9.326498061922570, −8.261177977090796, −7.738319187094316, −6.484511105266963, −4.874849179455167, −4.170842755611632, −2.175990025681517, 0, 2.175990025681517, 4.170842755611632, 4.874849179455167, 6.484511105266963, 7.738319187094316, 8.261177977090796, 9.326498061922570, 10.15928312968821, 10.96971784880108, 12.14616784601024, 12.81955495459951, 14.01653423556724, 14.63125822495919, 15.59156807507789, 16.60141841325293, 17.37589593635053, 17.79385769344014, 18.84623626477553, 19.51281592064860, 19.91606783402330

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