Properties

Degree 2
Conductor $ 3 \cdot 11 \cdot 41^{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 − 2·5-s + 6-s − 4·7-s − 3·8-s + 9-s − 2·10-s − 11-s − 12-s + 2·13-s − 4·14-s − 2·15-s − 16-s + 2·17-s + 18-s + 2·20-s − 4·21-s − 22-s + 8·23-s − 3·24-s − 25-s + 2·26-s + 27-s + 4·28-s + 6·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.51·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s + 0.554·13-s − 1.06·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.447·20-s − 0.872·21-s − 0.213·22-s + 1.66·23-s − 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.755·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55473 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55473 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(55473\)    =    \(3 \cdot 11 \cdot 41^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{55473} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 55473,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.400577193$
$L(\frac12)$  $\approx$  $1.400577193$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11,\;41\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 + T \)
41 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
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 \( 1 - 6 T + p T^{2} \)
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

−14.45723398115426, −13.78869977780645, −13.24248448459013, −12.99194751482921, −12.61803047111757, −12.06517581669402, −11.47319821389217, −10.86378119249553, −10.22956447033128, −9.577615076692339, −9.292155487232586, −8.720720661532326, −8.183236593273882, −7.600647724080528, −6.990712707815155, −6.387792898816592, −5.931635052959772, −5.118422031524717, −4.665912294805433, −3.824098741739433, −3.560656551849524, −3.059426293474153, −2.558003140395556, −1.233022849621807, −0.3817588866379413, 0.3817588866379413, 1.233022849621807, 2.558003140395556, 3.059426293474153, 3.560656551849524, 3.824098741739433, 4.665912294805433, 5.118422031524717, 5.931635052959772, 6.387792898816592, 6.990712707815155, 7.600647724080528, 8.183236593273882, 8.720720661532326, 9.292155487232586, 9.577615076692339, 10.22956447033128, 10.86378119249553, 11.47319821389217, 12.06517581669402, 12.61803047111757, 12.99194751482921, 13.24248448459013, 13.78869977780645, 14.45723398115426

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