Properties

Label 2-9680-1.1-c1-0-24
Degree $2$
Conductor $9680$
Sign $1$
Analytic cond. $77.2951$
Root an. cond. $8.79176$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.39·3-s − 5-s + 1.65·7-s − 1.05·9-s − 6.36·13-s + 1.39·15-s + 5.31·17-s − 4.36·19-s − 2.31·21-s + 5.57·23-s + 25-s + 5.65·27-s + 6.79·29-s − 0.259·31-s − 1.65·35-s − 0.791·37-s + 8.88·39-s + 2.74·41-s − 11.6·43-s + 1.05·45-s − 7.49·47-s − 4.25·49-s − 7.41·51-s − 1.84·53-s + 6.08·57-s + 7.15·59-s − 8.57·61-s + ⋯
L(s)  = 1  − 0.805·3-s − 0.447·5-s + 0.625·7-s − 0.350·9-s − 1.76·13-s + 0.360·15-s + 1.28·17-s − 1.00·19-s − 0.504·21-s + 1.16·23-s + 0.200·25-s + 1.08·27-s + 1.26·29-s − 0.0466·31-s − 0.279·35-s − 0.130·37-s + 1.42·39-s + 0.427·41-s − 1.77·43-s + 0.156·45-s − 1.09·47-s − 0.608·49-s − 1.03·51-s − 0.253·53-s + 0.806·57-s + 0.931·59-s − 1.09·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9680\)    =    \(2^{4} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(77.2951\)
Root analytic conductor: \(8.79176\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8850451813\)
\(L(\frac12)\) \(\approx\) \(0.8850451813\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
11 \( 1 \)
good3 \( 1 + 1.39T + 3T^{2} \)
7 \( 1 - 1.65T + 7T^{2} \)
13 \( 1 + 6.36T + 13T^{2} \)
17 \( 1 - 5.31T + 17T^{2} \)
19 \( 1 + 4.36T + 19T^{2} \)
23 \( 1 - 5.57T + 23T^{2} \)
29 \( 1 - 6.79T + 29T^{2} \)
31 \( 1 + 0.259T + 31T^{2} \)
37 \( 1 + 0.791T + 37T^{2} \)
41 \( 1 - 2.74T + 41T^{2} \)
43 \( 1 + 11.6T + 43T^{2} \)
47 \( 1 + 7.49T + 47T^{2} \)
53 \( 1 + 1.84T + 53T^{2} \)
59 \( 1 - 7.15T + 59T^{2} \)
61 \( 1 + 8.57T + 61T^{2} \)
67 \( 1 + 15.6T + 67T^{2} \)
71 \( 1 + 1.05T + 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 + 2.27T + 79T^{2} \)
83 \( 1 + 7.15T + 83T^{2} \)
89 \( 1 + 8.31T + 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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

−7.65722759468554121438223090141, −6.97079485222260087597193118328, −6.35601010890772905824442510880, −5.49483857237656444000423265018, −4.79935517826363527055032857700, −4.65939931831764024063808780571, −3.31061352988747627995593533045, −2.71046324923964394527678857508, −1.58115792541633857627551091148, −0.47152962199342511796424097722, 0.47152962199342511796424097722, 1.58115792541633857627551091148, 2.71046324923964394527678857508, 3.31061352988747627995593533045, 4.65939931831764024063808780571, 4.79935517826363527055032857700, 5.49483857237656444000423265018, 6.35601010890772905824442510880, 6.97079485222260087597193118328, 7.65722759468554121438223090141

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