Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{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  + 2·13-s + 6·17-s + 4·19-s + 6·23-s − 6·29-s + 4·31-s − 2·37-s + 6·41-s + 10·43-s + 6·47-s − 6·53-s + 12·59-s − 2·61-s − 2·67-s + 12·71-s + 2·73-s + 8·79-s − 6·83-s − 6·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.554·13-s + 1.45·17-s + 0.917·19-s + 1.25·23-s − 1.11·29-s + 0.718·31-s − 0.328·37-s + 0.937·41-s + 1.52·43-s + 0.875·47-s − 0.824·53-s + 1.56·59-s − 0.256·61-s − 0.244·67-s + 1.42·71-s + 0.234·73-s + 0.900·79-s − 0.658·83-s − 0.635·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(44100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{44100} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 44100,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.307569488$
$L(\frac12)$  $\approx$  $3.307569488$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
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

−14.60854895992694, −14.13553202257183, −13.75636077973061, −13.02158410809579, −12.67642573109773, −12.08668791301324, −11.56093277927159, −10.98942890112772, −10.61318429355468, −9.858266002301827, −9.428492608988158, −8.971962366843700, −8.253683703007742, −7.681877349377805, −7.297587982080817, −6.646114120208971, −5.838624716597478, −5.528856504674440, −4.906349726715783, −4.093601340408434, −3.501006815301042, −2.965553189500451, −2.196182336850910, −1.178246426729086, −0.7731435857637275, 0.7731435857637275, 1.178246426729086, 2.196182336850910, 2.965553189500451, 3.501006815301042, 4.093601340408434, 4.906349726715783, 5.528856504674440, 5.838624716597478, 6.646114120208971, 7.297587982080817, 7.681877349377805, 8.253683703007742, 8.971962366843700, 9.428492608988158, 9.858266002301827, 10.61318429355468, 10.98942890112772, 11.56093277927159, 12.08668791301324, 12.67642573109773, 13.02158410809579, 13.75636077973061, 14.13553202257183, 14.60854895992694

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