Properties

Label 2-9680-1.1-c1-0-3
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  − 0.220·3-s − 5-s − 2.08·7-s − 2.95·9-s − 1.87·13-s + 0.220·15-s − 5.28·17-s − 2.92·19-s + 0.461·21-s − 5.21·23-s + 25-s + 1.31·27-s + 3.81·29-s − 6.68·31-s + 2.08·35-s − 3.45·37-s + 0.414·39-s − 5.80·41-s + 0.884·43-s + 2.95·45-s − 7.69·47-s − 2.64·49-s + 1.16·51-s − 9.50·53-s + 0.645·57-s + 7.17·59-s + 7.88·61-s + ⋯
L(s)  = 1  − 0.127·3-s − 0.447·5-s − 0.789·7-s − 0.983·9-s − 0.521·13-s + 0.0570·15-s − 1.28·17-s − 0.670·19-s + 0.100·21-s − 1.08·23-s + 0.200·25-s + 0.252·27-s + 0.707·29-s − 1.20·31-s + 0.352·35-s − 0.568·37-s + 0.0664·39-s − 0.906·41-s + 0.134·43-s + 0.439·45-s − 1.12·47-s − 0.377·49-s + 0.163·51-s − 1.30·53-s + 0.0855·57-s + 0.934·59-s + 1.00·61-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

\(L(1)\) \(\approx\) \(0.1928352279\)
\(L(\frac12)\) \(\approx\) \(0.1928352279\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.220T + 3T^{2} \)
7 \( 1 + 2.08T + 7T^{2} \)
13 \( 1 + 1.87T + 13T^{2} \)
17 \( 1 + 5.28T + 17T^{2} \)
19 \( 1 + 2.92T + 19T^{2} \)
23 \( 1 + 5.21T + 23T^{2} \)
29 \( 1 - 3.81T + 29T^{2} \)
31 \( 1 + 6.68T + 31T^{2} \)
37 \( 1 + 3.45T + 37T^{2} \)
41 \( 1 + 5.80T + 41T^{2} \)
43 \( 1 - 0.884T + 43T^{2} \)
47 \( 1 + 7.69T + 47T^{2} \)
53 \( 1 + 9.50T + 53T^{2} \)
59 \( 1 - 7.17T + 59T^{2} \)
61 \( 1 - 7.88T + 61T^{2} \)
67 \( 1 - 1.32T + 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + 8.17T + 73T^{2} \)
79 \( 1 - 5.23T + 79T^{2} \)
83 \( 1 + 9.63T + 83T^{2} \)
89 \( 1 + 17.0T + 89T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−7.72153244174838282458428210814, −6.73224150340643714968017032438, −6.54543570350273937693352333148, −5.65024376291673207232899881637, −4.94467282496631936672362538991, −4.14353933855741929806823586690, −3.42488868643443628477085149059, −2.65387047928908913127220798179, −1.85593756457700494279467158999, −0.19539957511236389633654923597, 0.19539957511236389633654923597, 1.85593756457700494279467158999, 2.65387047928908913127220798179, 3.42488868643443628477085149059, 4.14353933855741929806823586690, 4.94467282496631936672362538991, 5.65024376291673207232899881637, 6.54543570350273937693352333148, 6.73224150340643714968017032438, 7.72153244174838282458428210814

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