Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 5^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 7·13-s − 7·19-s + 11·31-s − 10·37-s − 13·43-s − 6·49-s − 61-s + 11·67-s − 10·73-s − 4·79-s + 7·91-s − 19·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.94·13-s − 1.60·19-s + 1.97·31-s − 1.64·37-s − 1.98·43-s − 6/7·49-s − 0.128·61-s + 1.34·67-s − 1.17·73-s − 0.450·79-s + 0.733·91-s − 1.92·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{900} (1, \cdot )$
Sato-Tate  :  $N(\mathrm{U}(1))$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 900,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−19.90691116522799, −19.26619749102435, −18.99502813658477, −17.80384405755603, −17.14527859527019, −16.79210532294333, −15.70245868745332, −15.08875983334432, −14.44242568861775, −13.60142132405711, −12.71840331908473, −12.19470428841519, −11.39681998035065, −10.14453650340834, −10.01185297753130, −8.810318287138667, −8.068267125649044, −7.004186441705839, −6.413913812880263, −5.139585327725808, −4.421163301712993, −3.086011128891370, −2.050959770252986, 0, 2.050959770252986, 3.086011128891370, 4.421163301712993, 5.139585327725808, 6.413913812880263, 7.004186441705839, 8.068267125649044, 8.810318287138667, 10.01185297753130, 10.14453650340834, 11.39681998035065, 12.19470428841519, 12.71840331908473, 13.60142132405711, 14.44242568861775, 15.08875983334432, 15.70245868745332, 16.79210532294333, 17.14527859527019, 17.80384405755603, 18.99502813658477, 19.26619749102435, 19.90691116522799

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