Properties

Label 2-9680-1.1-c1-0-140
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  − 1.35·3-s + 5-s + 3.00·7-s − 1.17·9-s + 5.69·13-s − 1.35·15-s + 4.70·17-s + 3.69·19-s − 4.06·21-s + 6.83·23-s + 25-s + 5.64·27-s + 10.6·29-s + 3.50·31-s + 3.00·35-s + 5.79·37-s − 7.69·39-s + 8.39·41-s − 2.80·43-s − 1.17·45-s − 5.14·47-s + 2.04·49-s − 6.35·51-s − 12.2·53-s − 4.99·57-s − 3.30·59-s + 10.7·61-s + ⋯
L(s)  = 1  − 0.779·3-s + 0.447·5-s + 1.13·7-s − 0.391·9-s + 1.58·13-s − 0.348·15-s + 1.14·17-s + 0.848·19-s − 0.886·21-s + 1.42·23-s + 0.200·25-s + 1.08·27-s + 1.98·29-s + 0.629·31-s + 0.508·35-s + 0.952·37-s − 1.23·39-s + 1.31·41-s − 0.428·43-s − 0.175·45-s − 0.750·47-s + 0.291·49-s − 0.889·51-s − 1.67·53-s − 0.661·57-s − 0.430·59-s + 1.38·61-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

\(L(1)\) \(\approx\) \(2.709342080\)
\(L(\frac12)\) \(\approx\) \(2.709342080\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.35T + 3T^{2} \)
7 \( 1 - 3.00T + 7T^{2} \)
13 \( 1 - 5.69T + 13T^{2} \)
17 \( 1 - 4.70T + 17T^{2} \)
19 \( 1 - 3.69T + 19T^{2} \)
23 \( 1 - 6.83T + 23T^{2} \)
29 \( 1 - 10.6T + 29T^{2} \)
31 \( 1 - 3.50T + 31T^{2} \)
37 \( 1 - 5.79T + 37T^{2} \)
41 \( 1 - 8.39T + 41T^{2} \)
43 \( 1 + 2.80T + 43T^{2} \)
47 \( 1 + 5.14T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + 3.30T + 59T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 + 2.05T + 71T^{2} \)
73 \( 1 - 3.29T + 73T^{2} \)
79 \( 1 - 4.67T + 79T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 + 4.32T + 89T^{2} \)
97 \( 1 - 2.94T + 97T^{2} \)
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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

−7.967673559569418405984452472621, −6.75047496678155347327184458000, −6.27127487597631185003240413011, −5.58724116580104958295893792909, −5.04722076105367088556703387346, −4.47106539692592891815126923256, −3.30895275578112477486184098865, −2.70614501804200303072016384044, −1.19496943320718767919542530223, −1.09071032803553296888731433467, 1.09071032803553296888731433467, 1.19496943320718767919542530223, 2.70614501804200303072016384044, 3.30895275578112477486184098865, 4.47106539692592891815126923256, 5.04722076105367088556703387346, 5.58724116580104958295893792909, 6.27127487597631185003240413011, 6.75047496678155347327184458000, 7.967673559569418405984452472621

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