Properties

Label 2-9680-1.1-c1-0-26
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.87·3-s − 5-s − 4.15·7-s + 0.524·9-s − 2.27·13-s − 1.87·15-s − 2.80·17-s − 4.27·19-s − 7.80·21-s + 0.524·23-s + 25-s − 4.64·27-s + 3.75·29-s + 10.2·31-s + 4.15·35-s − 5.75·37-s − 4.27·39-s + 1.27·41-s − 6.96·43-s − 0.524·45-s + 10.4·47-s + 10.2·49-s − 5.26·51-s − 10.8·53-s − 8.03·57-s + 0.524·59-s + 8.47·61-s + ⋯
L(s)  = 1  + 1.08·3-s − 0.447·5-s − 1.57·7-s + 0.174·9-s − 0.632·13-s − 0.484·15-s − 0.680·17-s − 0.981·19-s − 1.70·21-s + 0.109·23-s + 0.200·25-s − 0.894·27-s + 0.697·29-s + 1.84·31-s + 0.702·35-s − 0.946·37-s − 0.685·39-s + 0.199·41-s − 1.06·43-s − 0.0782·45-s + 1.52·47-s + 1.46·49-s − 0.737·51-s − 1.48·53-s − 1.06·57-s + 0.0683·59-s + 1.08·61-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

\(L(1)\) \(\approx\) \(1.312137044\)
\(L(\frac12)\) \(\approx\) \(1.312137044\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.87T + 3T^{2} \)
7 \( 1 + 4.15T + 7T^{2} \)
13 \( 1 + 2.27T + 13T^{2} \)
17 \( 1 + 2.80T + 17T^{2} \)
19 \( 1 + 4.27T + 19T^{2} \)
23 \( 1 - 0.524T + 23T^{2} \)
29 \( 1 - 3.75T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + 5.75T + 37T^{2} \)
41 \( 1 - 1.27T + 41T^{2} \)
43 \( 1 + 6.96T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 - 0.524T + 59T^{2} \)
61 \( 1 - 8.47T + 61T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + 4.55T + 73T^{2} \)
79 \( 1 + 0.804T + 79T^{2} \)
83 \( 1 + 4.03T + 83T^{2} \)
89 \( 1 - 7.80T + 89T^{2} \)
97 \( 1 - 2.27T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.82409031487945673809017598716, −6.85747493547378048184039712012, −6.61438783936935823253767084192, −5.76366393338101357409219134698, −4.67929509433040781347683155397, −4.05761428393948340906839980039, −3.21785213906976482939815262609, −2.80580919538648282138548281871, −2.04708834894985636897804945219, −0.48009907031629176767734319500, 0.48009907031629176767734319500, 2.04708834894985636897804945219, 2.80580919538648282138548281871, 3.21785213906976482939815262609, 4.05761428393948340906839980039, 4.67929509433040781347683155397, 5.76366393338101357409219134698, 6.61438783936935823253767084192, 6.85747493547378048184039712012, 7.82409031487945673809017598716

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