Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11 $
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 + 4-s − 5-s − 8-s + 10-s + 11-s + 4·13-s + 16-s + 4·19-s − 20-s − 22-s + 25-s − 4·26-s + 6·29-s + 10·31-s − 32-s + 2·37-s − 4·38-s + 40-s − 12·41-s − 4·43-s + 44-s + 6·47-s − 50-s + 4·52-s + 6·53-s − 55-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 1.10·13-s + 1/4·16-s + 0.917·19-s − 0.223·20-s − 0.213·22-s + 1/5·25-s − 0.784·26-s + 1.11·29-s + 1.79·31-s − 0.176·32-s + 0.328·37-s − 0.648·38-s + 0.158·40-s − 1.87·41-s − 0.609·43-s + 0.150·44-s + 0.875·47-s − 0.141·50-s + 0.554·52-s + 0.824·53-s − 0.134·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(48510\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{48510} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 48510,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.970764274$
$L(\frac12)$  $\approx$  $1.970764274$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 - T \)
good13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.73243454714719, −13.85541306379040, −13.65513688549181, −13.12472024719323, −12.13140697946173, −11.92418156919625, −11.60091427289025, −10.79962884082518, −10.40150877818380, −9.948712452515610, −9.228691705972424, −8.774731848325377, −8.226500745967112, −7.889002972016943, −7.152164220076175, −6.546785741503765, −6.238125266568608, −5.409163192250150, −4.760614641894568, −4.084141772698620, −3.335975859856841, −2.934270756636977, −1.978372019403274, −1.151096643130589, −0.6528083935339522, 0.6528083935339522, 1.151096643130589, 1.978372019403274, 2.934270756636977, 3.335975859856841, 4.084141772698620, 4.760614641894568, 5.409163192250150, 6.238125266568608, 6.546785741503765, 7.152164220076175, 7.889002972016943, 8.226500745967112, 8.774731848325377, 9.228691705972424, 9.948712452515610, 10.40150877818380, 10.79962884082518, 11.60091427289025, 11.92418156919625, 12.13140697946173, 13.12472024719323, 13.65513688549181, 13.85541306379040, 14.73243454714719

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