Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 12-s − 13-s + 15-s + 16-s − 6·17-s + 18-s + 4·19-s − 20-s − 24-s + 25-s − 26-s − 27-s + 6·29-s + 30-s + 4·31-s + 32-s − 6·34-s + 36-s − 10·37-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-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 − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 1.11·29-s + 0.182·30-s + 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 1.64·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(19110\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{19110} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 19110,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 \)
13 \( 1 + T \)
good11 \( 1 + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−15.84859603805379, −15.55039245021857, −15.01997393161103, −14.28065920268582, −13.66938462336493, −13.42240938128020, −12.52544418757404, −12.15587596082650, −11.73763216673486, −11.09575453303006, −10.62385907277813, −10.06576402265888, −9.279157651832289, −8.639653716621920, −7.953281542956160, −7.247207196629511, −6.740282845110125, −6.254531905950239, −5.432131022193840, −4.833446898020056, −4.414652588886978, −3.610213300536440, −2.898753781437420, −2.084886363620472, −1.102796947237552, 0, 1.102796947237552, 2.084886363620472, 2.898753781437420, 3.610213300536440, 4.414652588886978, 4.833446898020056, 5.432131022193840, 6.254531905950239, 6.740282845110125, 7.247207196629511, 7.953281542956160, 8.639653716621920, 9.279157651832289, 10.06576402265888, 10.62385907277813, 11.09575453303006, 11.73763216673486, 12.15587596082650, 12.52544418757404, 13.42240938128020, 13.66938462336493, 14.28065920268582, 15.01997393161103, 15.55039245021857, 15.84859603805379

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