Properties

Degree $2$
Conductor $9408$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

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

Degree: \(2\)
Conductor: \(9408\)    =    \(2^{6} \cdot 3 \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{9408} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9408,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
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
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.30576192335939904214429203550, −6.87092532125202661203160267877, −6.28152140369262438285565225280, −4.98962289326104261798410227186, −4.39134080888169013303153690359, −3.83990440919322935449915464847, −3.22989944011303972750872174630, −2.26021299805192728319195793621, −1.18709860967949250865613860532, 0, 1.18709860967949250865613860532, 2.26021299805192728319195793621, 3.22989944011303972750872174630, 3.83990440919322935449915464847, 4.39134080888169013303153690359, 4.98962289326104261798410227186, 6.28152140369262438285565225280, 6.87092532125202661203160267877, 7.30576192335939904214429203550

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