Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7^{2} \cdot 11 $
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 − 3-s − 4-s + 6-s + 3·8-s + 9-s + 11-s + 12-s − 2·13-s − 16-s − 2·17-s − 18-s − 22-s − 8·23-s − 3·24-s + 2·26-s − 27-s − 6·29-s + 8·31-s − 5·32-s − 33-s + 2·34-s − 36-s − 6·37-s + 2·39-s + 2·41-s − 44-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 0.554·13-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.213·22-s − 1.66·23-s − 0.612·24-s + 0.392·26-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.883·32-s − 0.174·33-s + 0.342·34-s − 1/6·36-s − 0.986·37-s + 0.320·39-s + 0.312·41-s − 0.150·44-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 40425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 40425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(40425\)    =    \(3 \cdot 5^{2} \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{40425} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 40425,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−15.06839898370076, −14.46063913636934, −13.91944057038330, −13.54946466409561, −12.92669765683386, −12.31588986725397, −11.92105720908832, −11.32496002311413, −10.70134183724355, −10.19194322579723, −9.836340510925709, −9.232282622814423, −8.736492542686742, −8.127087016923967, −7.563814822849048, −7.122556843295325, −6.324674917374737, −5.840086653096625, −5.135179027316095, −4.493608447860747, −4.107679977271536, −3.331815423518953, −2.233990191375352, −1.683190300847068, −0.7262044861124459, 0, 0.7262044861124459, 1.683190300847068, 2.233990191375352, 3.331815423518953, 4.107679977271536, 4.493608447860747, 5.135179027316095, 5.840086653096625, 6.324674917374737, 7.122556843295325, 7.563814822849048, 8.127087016923967, 8.736492542686742, 9.232282622814423, 9.836340510925709, 10.19194322579723, 10.70134183724355, 11.32496002311413, 11.92105720908832, 12.31588986725397, 12.92669765683386, 13.54946466409561, 13.91944057038330, 14.46063913636934, 15.06839898370076

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