Properties

Degree 2
Conductor $ 3^{2} \cdot 239 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 5-s − 1.41i·7-s + 8-s − 10-s + 11-s + 1.41i·13-s + 1.41i·14-s − 16-s + 17-s + 1.41i·19-s − 22-s + 1.41i·23-s − 1.41i·26-s + 29-s + ⋯
L(s)  = 1  − 2-s + 5-s − 1.41i·7-s + 8-s − 10-s + 11-s + 1.41i·13-s + 1.41i·14-s − 16-s + 17-s + 1.41i·19-s − 22-s + 1.41i·23-s − 1.41i·26-s + 29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2151\)    =    \(3^{2} \cdot 239\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{2151} (955, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2151,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.8490923334\)
\(L(\frac12)\)  \(\approx\)  \(0.8490923334\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;239\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;239\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
239 \( 1 - T \)
good2 \( 1 + T + T^{2} \)
5 \( 1 - T + T^{2} \)
7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + 1.41iT - T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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

−9.403900991727383454971185836199, −8.735792119741634860765562326784, −7.58930044922002904632021011796, −7.26809148398465079991793981317, −6.28936765790551897519371521509, −5.42078580743045591351431096127, −4.15390296872720246988521497798, −3.75070727266357459269629251941, −1.78027265371229092511307052832, −1.28936431270954069317179779520, 1.07026718400937328860511981678, 2.25127978319250414000309853719, 3.12570361017273950383304364602, 4.67576576351142655721747271211, 5.38510191980781384452550227640, 6.16191576010009351777978057046, 6.96484129443832487962415621855, 8.134508540993616110825691166980, 8.613284827665564718178877719368, 9.208704971631530789653962510550

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