Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 \cdot 11^{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  + 2-s − 3-s − 4-s − 6-s + 7-s − 3·8-s + 9-s + 12-s − 6·13-s + 14-s − 16-s + 2·17-s + 18-s + 8·19-s − 21-s − 8·23-s + 3·24-s − 6·26-s − 27-s − 28-s + 2·29-s + 4·31-s + 5·32-s + 2·34-s − 36-s + 2·37-s + 8·38-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.288·12-s − 1.66·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 1.83·19-s − 0.218·21-s − 1.66·23-s + 0.612·24-s − 1.17·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + 0.718·31-s + 0.883·32-s + 0.342·34-s − 1/6·36-s + 0.328·37-s + 1.29·38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 63525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 63525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(63525\)    =    \(3 \cdot 5^{2} \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{63525} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 63525,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good2 \( 1 - T + p T^{2} \)
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

−14.41176994506969, −14.10989360222402, −13.55409729812714, −12.92525057968236, −12.43320022047063, −12.00941989372019, −11.70262748697354, −11.24659858256940, −10.18469896148672, −9.984825009352070, −9.584005381837561, −8.967055866667385, −8.168582457912486, −7.646728369464692, −7.350409640866583, −6.433859503458820, −5.919236810693479, −5.431165490993412, −4.906504501802016, −4.511668319539569, −3.933265169347224, −3.091344533097303, −2.660088166790085, −1.688225576602508, −0.8132522041137054, 0, 0.8132522041137054, 1.688225576602508, 2.660088166790085, 3.091344533097303, 3.933265169347224, 4.511668319539569, 4.906504501802016, 5.431165490993412, 5.919236810693479, 6.433859503458820, 7.350409640866583, 7.646728369464692, 8.168582457912486, 8.967055866667385, 9.584005381837561, 9.984825009352070, 10.18469896148672, 11.24659858256940, 11.70262748697354, 12.00941989372019, 12.43320022047063, 12.92525057968236, 13.55409729812714, 14.10989360222402, 14.41176994506969

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