↑

Properties

Label 2-12e3-3.2-c0-0-1
Degree $2$
Conductor $1728$
Sign $1$
Analytic cond. $0.862384$
Root an. cond. $0.928646$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 13-s − 19-s + 25-s − 2·31-s + 37-s + 2·43-s + 61-s − 67-s − 73-s + 79-s + 91-s − 97-s + 103-s − 2·109-s + ⋯
L(s)  = 1  + 7-s + 13-s − 19-s + 25-s − 2·31-s + 37-s + 2·43-s + 61-s − 67-s − 73-s + 79-s + 91-s − 97-s + 103-s − 2·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1728} (1025, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.295060619\)
\(L(\frac12)\) \(\approx\) \(1.295060619\)
\(L(1)\) 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 \( ( 1 - T )( 1 + T ) \)
7 \( 1 - T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 - T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 + T )^{2} \)
37 \( 1 - T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 - T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
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

−9.293231364519817793419810254168, −8.738561855671784908390292831238, −7.989250395116094548223721998903, −7.22334147578833343759924712669, −6.24409988372751349996364927770, −5.47025385860495809217322035139, −4.51241628285495777414575770935, −3.74308983229530032736095950668, −2.43982583189414567843494815700, −1.33132348340000374378114128376, 1.33132348340000374378114128376, 2.43982583189414567843494815700, 3.74308983229530032736095950668, 4.51241628285495777414575770935, 5.47025385860495809217322035139, 6.24409988372751349996364927770, 7.22334147578833343759924712669, 7.989250395116094548223721998903, 8.738561855671784908390292831238, 9.293231364519817793419810254168

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