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 0

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 − 4·19-s + 21-s + 3·24-s − 6·26-s + 27-s − 28-s + 2·29-s + 8·31-s − 5·32-s − 2·34-s − 36-s − 6·37-s + 4·38-s + 6·39-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 − 0.917·19-s + 0.218·21-s + 0.612·24-s − 1.17·26-s + 0.192·27-s − 0.188·28-s + 0.371·29-s + 1.43·31-s − 0.883·32-s − 0.342·34-s − 1/6·36-s − 0.986·37-s + 0.648·38-s + 0.960·39-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  =  0
Selberg data  =  $(2,\ 63525,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.132463656$
$L(\frac12)$  $\approx$  $2.132463656$
$L(\frac{3}{2})$   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 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−14.26189372277564, −13.66289229110031, −13.38091797802903, −12.94318175322065, −12.19603936752720, −11.70234628051949, −11.02404778796694, −10.48509119763851, −10.21366571053964, −9.602548796626016, −8.929499305709319, −8.582400670605382, −8.144968901476265, −7.950134593650501, −6.886780432685989, −6.672570346919307, −5.809682209674020, −5.166761254784251, −4.590378920767033, −3.881092952220018, −3.583583708273784, −2.688870968314585, −1.822725792999675, −1.302213481603797, −0.5917745804792084, 0.5917745804792084, 1.302213481603797, 1.822725792999675, 2.688870968314585, 3.583583708273784, 3.881092952220018, 4.590378920767033, 5.166761254784251, 5.809682209674020, 6.672570346919307, 6.886780432685989, 7.950134593650501, 8.144968901476265, 8.582400670605382, 8.929499305709319, 9.602548796626016, 10.21366571053964, 10.48509119763851, 11.02404778796694, 11.70234628051949, 12.19603936752720, 12.94318175322065, 13.38091797802903, 13.66289229110031, 14.26189372277564

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