Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 349 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s − 2·11-s + 12-s + 14-s + 15-s + 16-s − 3·17-s + 18-s + 6·19-s + 20-s + 21-s − 2·22-s − 4·23-s + 24-s + 25-s + 27-s + 28-s + 9·29-s + 30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.218·21-s − 0.426·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.188·28-s + 1.67·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 10470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 10470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10470\)    =    \(2 \cdot 3 \cdot 5 \cdot 349\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{10470} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 10470,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.144842188$
$L(\frac12)$  $\approx$  $5.144842188$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;349\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;349\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
349 \( 1 + T \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.25140132303788, −15.91555181726849, −15.42527579974137, −14.72138105471962, −14.18508709122182, −13.73739911295227, −13.28894970927238, −12.74669842720010, −11.79439857202693, −11.68597990567819, −10.69511596184891, −10.07322522551455, −9.729577133498347, −8.720925617950207, −8.235876937092878, −7.616880145610547, −6.836057718900128, −6.301911574684966, −5.396254886178816, −4.912478026629144, −4.202413336871992, −3.322835997379591, −2.635959717588869, −1.991919750467729, −0.9494431811259593, 0.9494431811259593, 1.991919750467729, 2.635959717588869, 3.322835997379591, 4.202413336871992, 4.912478026629144, 5.396254886178816, 6.301911574684966, 6.836057718900128, 7.616880145610547, 8.235876937092878, 8.720925617950207, 9.729577133498347, 10.07322522551455, 10.69511596184891, 11.68597990567819, 11.79439857202693, 12.74669842720010, 13.28894970927238, 13.73739911295227, 14.18508709122182, 14.72138105471962, 15.42527579974137, 15.91555181726849, 16.25140132303788

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