↑

Properties

Label 2-432-4.3-c0-0-1
Degree $2$
Conductor $432$
Sign $0.866 + 0.5i$
Analytic cond. $0.215596$
Root an. cond. $0.464323$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·7-s + 13-s + 1.73i·19-s − 25-s − 37-s − 1.99·49-s − 61-s + 1.73i·67-s + 73-s + 1.73i·79-s − 1.73i·91-s + 97-s − 1.73i·103-s − 2·109-s + ⋯
L(s)  = 1  − 1.73i·7-s + 13-s + 1.73i·19-s − 25-s − 37-s − 1.99·49-s − 61-s + 1.73i·67-s + 73-s + 1.73i·79-s − 1.73i·91-s + 97-s − 1.73i·103-s − 2·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(0.215596\)
Root analytic conductor: \(0.464323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :0),\ 0.866 + 0.5i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8598529884\)
\(L(\frac12)\) \(\approx\) \(0.8598529884\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + T^{2} \)
7 \( 1 + 1.73iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 1.73iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - 1.73iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - 1.73iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T + T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.11506384040874599174575530778, −10.41318810156162890993105053024, −9.741458735852807844804829277372, −8.389909325426662365467937910448, −7.65038060218050400329589679060, −6.69847889097025432597841092210, −5.66564954745531142939780264109, −4.18113726594031045805368273925, −3.55799567179046526794633837321, −1.46097974006279403355988180347, 2.09526533679096985646765972830, 3.28411205647023278960401046928, 4.82881851463021965946776033904, 5.78000947512797949857156473419, 6.61092543395892363240057783301, 7.956187752910641576281640228384, 8.902403920509623737121451616051, 9.333613494056748212348670577076, 10.68984054788385981715659358489, 11.53775779762523206960718733194

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