Properties

Label 2-9680-1.1-c1-0-119
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  + 2.19·3-s + 5-s + 0.544·7-s + 1.81·9-s + 1.63·13-s + 2.19·15-s + 0.169·17-s + 4.72·19-s + 1.19·21-s + 2.26·23-s + 25-s − 2.60·27-s + 9.43·29-s + 3.01·31-s + 0.544·35-s − 0.0537·37-s + 3.59·39-s − 2.54·41-s + 7.46·43-s + 1.81·45-s − 9.07·47-s − 6.70·49-s + 0.370·51-s − 2.03·53-s + 10.3·57-s + 12.6·59-s + 2.80·61-s + ⋯
L(s)  = 1  + 1.26·3-s + 0.447·5-s + 0.205·7-s + 0.603·9-s + 0.454·13-s + 0.566·15-s + 0.0409·17-s + 1.08·19-s + 0.260·21-s + 0.471·23-s + 0.200·25-s − 0.501·27-s + 1.75·29-s + 0.540·31-s + 0.0919·35-s − 0.00882·37-s + 0.575·39-s − 0.397·41-s + 1.13·43-s + 0.270·45-s − 1.32·47-s − 0.957·49-s + 0.0519·51-s − 0.279·53-s + 1.37·57-s + 1.65·59-s + 0.359·61-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

\(L(1)\) \(\approx\) \(4.288300297\)
\(L(\frac12)\) \(\approx\) \(4.288300297\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.19T + 3T^{2} \)
7 \( 1 - 0.544T + 7T^{2} \)
13 \( 1 - 1.63T + 13T^{2} \)
17 \( 1 - 0.169T + 17T^{2} \)
19 \( 1 - 4.72T + 19T^{2} \)
23 \( 1 - 2.26T + 23T^{2} \)
29 \( 1 - 9.43T + 29T^{2} \)
31 \( 1 - 3.01T + 31T^{2} \)
37 \( 1 + 0.0537T + 37T^{2} \)
41 \( 1 + 2.54T + 41T^{2} \)
43 \( 1 - 7.46T + 43T^{2} \)
47 \( 1 + 9.07T + 47T^{2} \)
53 \( 1 + 2.03T + 53T^{2} \)
59 \( 1 - 12.6T + 59T^{2} \)
61 \( 1 - 2.80T + 61T^{2} \)
67 \( 1 - 1.26T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 - 7.97T + 73T^{2} \)
79 \( 1 + 3.13T + 79T^{2} \)
83 \( 1 + 2.73T + 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 - 7.89T + 97T^{2} \)
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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

−7.85573360902701250730206263459, −7.08669443138321823485192076528, −6.43537990784872008196409833332, −5.59813905902168839793310999836, −4.87569400824977694059254090420, −4.07420728462973788136728504691, −3.14725577371695867566708786319, −2.80292111403127588744266631508, −1.82714445257349625185965010198, −0.975872508083668805537476205201, 0.975872508083668805537476205201, 1.82714445257349625185965010198, 2.80292111403127588744266631508, 3.14725577371695867566708786319, 4.07420728462973788136728504691, 4.87569400824977694059254090420, 5.59813905902168839793310999836, 6.43537990784872008196409833332, 7.08669443138321823485192076528, 7.85573360902701250730206263459

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