Properties

Label 2-9664-1.1-c1-0-93
Degree $2$
Conductor $9664$
Sign $1$
Analytic cond. $77.1674$
Root an. cond. $8.78449$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·7-s − 2·9-s + 2·11-s − 2·13-s − 5·17-s + 4·19-s + 2·21-s + 6·23-s − 5·25-s − 5·27-s − 3·31-s + 2·33-s + 2·37-s − 2·39-s − 4·41-s + 6·43-s + 9·47-s − 3·49-s − 5·51-s − 53-s + 4·57-s + 6·59-s + 13·61-s − 4·63-s + 15·67-s + 6·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s − 2/3·9-s + 0.603·11-s − 0.554·13-s − 1.21·17-s + 0.917·19-s + 0.436·21-s + 1.25·23-s − 25-s − 0.962·27-s − 0.538·31-s + 0.348·33-s + 0.328·37-s − 0.320·39-s − 0.624·41-s + 0.914·43-s + 1.31·47-s − 3/7·49-s − 0.700·51-s − 0.137·53-s + 0.529·57-s + 0.781·59-s + 1.66·61-s − 0.503·63-s + 1.83·67-s + 0.722·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9664\)    =    \(2^{6} \cdot 151\)
Sign: $1$
Analytic conductor: \(77.1674\)
Root analytic conductor: \(8.78449\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9664,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.556921677\)
\(L(\frac12)\) \(\approx\) \(2.556921677\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
151 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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

−7.60021948092764537632079419322, −7.21899013653519796570120960038, −6.36071595459821884962031025332, −5.52281653013098876677624717854, −4.96377522191538833077622427597, −4.12507235348094948962973805824, −3.42411867458366080974259746990, −2.50599320209037879567578672909, −1.92447861180603306823598813116, −0.73169470468344138877721610044, 0.73169470468344138877721610044, 1.92447861180603306823598813116, 2.50599320209037879567578672909, 3.42411867458366080974259746990, 4.12507235348094948962973805824, 4.96377522191538833077622427597, 5.52281653013098876677624717854, 6.36071595459821884962031025332, 7.21899013653519796570120960038, 7.60021948092764537632079419322

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