Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 \cdot 17^{2} $
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 − 2·5-s − 6-s + 2·7-s − 8-s + 9-s + 2·10-s − 11-s + 12-s + 3·13-s − 2·14-s − 2·15-s + 16-s − 18-s − 5·19-s − 2·20-s + 2·21-s + 22-s − 3·23-s − 24-s − 25-s − 3·26-s + 27-s + 2·28-s − 5·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s + 0.832·13-s − 0.534·14-s − 0.516·15-s + 1/4·16-s − 0.235·18-s − 1.14·19-s − 0.447·20-s + 0.436·21-s + 0.213·22-s − 0.625·23-s − 0.204·24-s − 1/5·25-s − 0.588·26-s + 0.192·27-s + 0.377·28-s − 0.928·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(19074\)    =    \(2 \cdot 3 \cdot 11 \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{19074} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 19074,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.498940706\)
\(L(\frac12)\)  \(\approx\)  \(1.498940706\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
11 \( 1 + T \)
17 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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

−15.68934987972060, −15.32141038915723, −14.72419381348885, −14.29297568147171, −13.43425972241392, −13.11751186327650, −12.23832845982676, −11.77218707986306, −11.21591965394307, −10.77821030622227, −10.08549443909901, −9.585592720934379, −8.614588156561750, −8.416563749893350, −7.995420861042288, −7.398333618661327, −6.718336196221820, −6.033267446479360, −5.251062724198204, −4.199655104337186, −4.039851118914975, −3.052619440326884, −2.268153286788182, −1.561161238890308, −0.5586918947628366, 0.5586918947628366, 1.561161238890308, 2.268153286788182, 3.052619440326884, 4.039851118914975, 4.199655104337186, 5.251062724198204, 6.033267446479360, 6.718336196221820, 7.398333618661327, 7.995420861042288, 8.416563749893350, 8.614588156561750, 9.585592720934379, 10.08549443909901, 10.77821030622227, 11.21591965394307, 11.77218707986306, 12.23832845982676, 13.11751186327650, 13.43425972241392, 14.29297568147171, 14.72419381348885, 15.32141038915723, 15.68934987972060

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