Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 11 \cdot 29^{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  − 3-s + 2·5-s − 2·7-s + 9-s − 11-s − 2·13-s − 2·15-s − 4·17-s + 6·19-s + 2·21-s − 25-s − 27-s + 8·31-s + 33-s − 4·35-s − 10·37-s + 2·39-s − 8·41-s + 2·43-s + 2·45-s + 8·47-s − 3·49-s + 4·51-s − 2·53-s − 2·55-s − 6·57-s + 12·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.516·15-s − 0.970·17-s + 1.37·19-s + 0.436·21-s − 1/5·25-s − 0.192·27-s + 1.43·31-s + 0.174·33-s − 0.676·35-s − 1.64·37-s + 0.320·39-s − 1.24·41-s + 0.304·43-s + 0.298·45-s + 1.16·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s − 0.269·55-s − 0.794·57-s + 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(111012\)    =    \(2^{2} \cdot 3 \cdot 11 \cdot 29^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{111012} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 111012,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.479803823$
$L(\frac12)$  $\approx$  $1.479803823$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;29\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
11 \( 1 + T \)
29 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−13.56109034095594, −13.29517151723232, −12.69213393911131, −12.02346730295794, −11.93325035714968, −11.21200459568841, −10.60272393316834, −10.11373484222034, −9.899048692944655, −9.290112409962489, −8.914992816386997, −8.171577575189203, −7.577495998518472, −6.978842254908827, −6.557143885778139, −6.153353960847992, −5.479073536625487, −5.078239930767316, −4.646073384599693, −3.713230099858039, −3.275077229230644, −2.441189916012835, −2.061460683420310, −1.172425415975873, −0.4096804924984696, 0.4096804924984696, 1.172425415975873, 2.061460683420310, 2.441189916012835, 3.275077229230644, 3.713230099858039, 4.646073384599693, 5.078239930767316, 5.479073536625487, 6.153353960847992, 6.557143885778139, 6.978842254908827, 7.577495998518472, 8.171577575189203, 8.914992816386997, 9.290112409962489, 9.899048692944655, 10.11373484222034, 10.60272393316834, 11.21200459568841, 11.93325035714968, 12.02346730295794, 12.69213393911131, 13.29517151723232, 13.56109034095594

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