Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 43^{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 + 5-s − 6-s + 4·7-s + 8-s + 9-s + 10-s − 12-s + 2·13-s + 4·14-s − 15-s + 16-s + 6·17-s + 18-s + 4·19-s + 20-s − 4·21-s − 24-s + 25-s + 2·26-s − 27-s + 4·28-s + 6·29-s − 30-s + 8·31-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.554·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.872·21-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.755·28-s + 1.11·29-s − 0.182·30-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(55470\)    =    \(2 \cdot 3 \cdot 5 \cdot 43^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{55470} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 55470,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(6.359513070\)
\(L(\frac12)\)  \(\approx\)  \(6.359513070\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 - T \)
43 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.30175509093558, −13.85963091804444, −13.62197747665921, −12.85684376822419, −12.15772849115649, −11.95322123453312, −11.45883909693719, −10.97293613990074, −10.31339557709329, −10.05059985148236, −9.341737973443094, −8.396324625445847, −8.176363400816943, −7.542667296108351, −6.933107218597967, −6.292594888535864, −5.776966035779799, −5.140767037115811, −4.948518269033866, −4.275468408972610, −3.491184913195611, −2.897073090378688, −2.027393466319607, −1.287033357223867, −0.9384267492322601, 0.9384267492322601, 1.287033357223867, 2.027393466319607, 2.897073090378688, 3.491184913195611, 4.275468408972610, 4.948518269033866, 5.140767037115811, 5.776966035779799, 6.292594888535864, 6.933107218597967, 7.542667296108351, 8.176363400816943, 8.396324625445847, 9.341737973443094, 10.05059985148236, 10.31339557709329, 10.97293613990074, 11.45883909693719, 11.95322123453312, 12.15772849115649, 12.85684376822419, 13.62197747665921, 13.85963091804444, 14.30175509093558

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