Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 7^{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 + 3.41·5-s + 9-s − 4.82·11-s + 1.41·13-s + 3.41·15-s + 6.24·17-s + 1.17·19-s − 0.828·23-s + 6.65·25-s + 27-s + 8.48·29-s − 10.8·31-s − 4.82·33-s + 9.65·37-s + 1.41·39-s + 3.41·41-s + 8·43-s + 3.41·45-s − 1.17·47-s + 6.24·51-s − 9.31·53-s − 16.4·55-s + 1.17·57-s + 10.8·59-s − 5.89·61-s + 4.82·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.52·5-s + 0.333·9-s − 1.45·11-s + 0.392·13-s + 0.881·15-s + 1.51·17-s + 0.268·19-s − 0.172·23-s + 1.33·25-s + 0.192·27-s + 1.57·29-s − 1.94·31-s − 0.840·33-s + 1.58·37-s + 0.226·39-s + 0.533·41-s + 1.21·43-s + 0.508·45-s − 0.170·47-s + 0.874·51-s − 1.27·53-s − 2.22·55-s + 0.155·57-s + 1.40·59-s − 0.755·61-s + 0.598·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9408\)    =    \(2^{6} \cdot 3 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{9408} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 9408,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.800323547\)
\(L(\frac12)\)  \(\approx\)  \(3.800323547\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 \)
good5 \( 1 - 3.41T + 5T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
17 \( 1 - 6.24T + 17T^{2} \)
19 \( 1 - 1.17T + 19T^{2} \)
23 \( 1 + 0.828T + 23T^{2} \)
29 \( 1 - 8.48T + 29T^{2} \)
31 \( 1 + 10.8T + 31T^{2} \)
37 \( 1 - 9.65T + 37T^{2} \)
41 \( 1 - 3.41T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 1.17T + 47T^{2} \)
53 \( 1 + 9.31T + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 + 5.89T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 - 4.82T + 71T^{2} \)
73 \( 1 + 3.07T + 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 + 7.31T + 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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

−7.76535444669679082441199973285, −7.13191183770128123403107569857, −6.14316828410028836453823281925, −5.65193519067449950241227579217, −5.16910165833190938267959287824, −4.21363644611502808354321523125, −3.11713948337185822678921272673, −2.66197967867243150344768125470, −1.84391093301018242056777715998, −0.940470475587217812319300795726, 0.940470475587217812319300795726, 1.84391093301018242056777715998, 2.66197967867243150344768125470, 3.11713948337185822678921272673, 4.21363644611502808354321523125, 5.16910165833190938267959287824, 5.65193519067449950241227579217, 6.14316828410028836453823281925, 7.13191183770128123403107569857, 7.76535444669679082441199973285

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