Properties

Label 2-9680-1.1-c1-0-114
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  + 2.96·3-s + 5-s − 1.02·7-s + 5.79·9-s − 0.504·13-s + 2.96·15-s − 3.93·17-s − 2.50·19-s − 3.05·21-s + 5.73·23-s + 25-s + 8.29·27-s + 6.92·29-s − 4.47·31-s − 1.02·35-s + 4.78·37-s − 1.49·39-s + 11.8·41-s + 7.10·43-s + 5.79·45-s + 0.182·47-s − 5.94·49-s − 11.6·51-s + 12.4·53-s − 7.42·57-s + 10.5·59-s − 10.9·61-s + ⋯
L(s)  = 1  + 1.71·3-s + 0.447·5-s − 0.388·7-s + 1.93·9-s − 0.139·13-s + 0.765·15-s − 0.953·17-s − 0.574·19-s − 0.665·21-s + 1.19·23-s + 0.200·25-s + 1.59·27-s + 1.28·29-s − 0.804·31-s − 0.173·35-s + 0.786·37-s − 0.239·39-s + 1.84·41-s + 1.08·43-s + 0.864·45-s + 0.0266·47-s − 0.848·49-s − 1.63·51-s + 1.71·53-s − 0.983·57-s + 1.37·59-s − 1.39·61-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

\(L(1)\) \(\approx\) \(4.493972820\)
\(L(\frac12)\) \(\approx\) \(4.493972820\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.96T + 3T^{2} \)
7 \( 1 + 1.02T + 7T^{2} \)
13 \( 1 + 0.504T + 13T^{2} \)
17 \( 1 + 3.93T + 17T^{2} \)
19 \( 1 + 2.50T + 19T^{2} \)
23 \( 1 - 5.73T + 23T^{2} \)
29 \( 1 - 6.92T + 29T^{2} \)
31 \( 1 + 4.47T + 31T^{2} \)
37 \( 1 - 4.78T + 37T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 - 7.10T + 43T^{2} \)
47 \( 1 - 0.182T + 47T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 + 9.76T + 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 + 7.32T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + 4.33T + 83T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 + 7.80T + 97T^{2} \)
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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

−7.69280483394025538941030308527, −7.15525763274346310832521716214, −6.50641807562197386067609789712, −5.71242944831212386746234234429, −4.59258473456345985034540037943, −4.16120257503450722960771486372, −3.19868764383873973736334668270, −2.60656956313876762760740627896, −2.07657950512240448532551952136, −0.938787949846282974913576350891, 0.938787949846282974913576350891, 2.07657950512240448532551952136, 2.60656956313876762760740627896, 3.19868764383873973736334668270, 4.16120257503450722960771486372, 4.59258473456345985034540037943, 5.71242944831212386746234234429, 6.50641807562197386067609789712, 7.15525763274346310832521716214, 7.69280483394025538941030308527

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