Properties

Label 2-9680-1.1-c1-0-33
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  − 2.34·3-s − 5-s − 2.86·7-s + 2.51·9-s + 6.24·13-s + 2.34·15-s − 3.73·17-s + 8.24·19-s + 6.73·21-s − 8.94·23-s + 25-s + 1.13·27-s + 8.69·29-s + 5.21·31-s + 2.86·35-s − 2.69·37-s − 14.6·39-s + 8.21·41-s − 7.13·43-s − 2.51·45-s − 1.31·47-s + 1.21·49-s + 8.76·51-s − 0.180·53-s − 19.3·57-s − 3.55·59-s + 5.94·61-s + ⋯
L(s)  = 1  − 1.35·3-s − 0.447·5-s − 1.08·7-s + 0.839·9-s + 1.73·13-s + 0.606·15-s − 0.905·17-s + 1.89·19-s + 1.46·21-s − 1.86·23-s + 0.200·25-s + 0.218·27-s + 1.61·29-s + 0.936·31-s + 0.484·35-s − 0.443·37-s − 2.35·39-s + 1.28·41-s − 1.08·43-s − 0.375·45-s − 0.191·47-s + 0.173·49-s + 1.22·51-s − 0.0247·53-s − 2.56·57-s − 0.462·59-s + 0.761·61-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

\(L(1)\) \(\approx\) \(0.8270151449\)
\(L(\frac12)\) \(\approx\) \(0.8270151449\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.34T + 3T^{2} \)
7 \( 1 + 2.86T + 7T^{2} \)
13 \( 1 - 6.24T + 13T^{2} \)
17 \( 1 + 3.73T + 17T^{2} \)
19 \( 1 - 8.24T + 19T^{2} \)
23 \( 1 + 8.94T + 23T^{2} \)
29 \( 1 - 8.69T + 29T^{2} \)
31 \( 1 - 5.21T + 31T^{2} \)
37 \( 1 + 2.69T + 37T^{2} \)
41 \( 1 - 8.21T + 41T^{2} \)
43 \( 1 + 7.13T + 43T^{2} \)
47 \( 1 + 1.31T + 47T^{2} \)
53 \( 1 + 0.180T + 53T^{2} \)
59 \( 1 + 3.55T + 59T^{2} \)
61 \( 1 - 5.94T + 61T^{2} \)
67 \( 1 + 2.27T + 67T^{2} \)
71 \( 1 - 2.51T + 71T^{2} \)
73 \( 1 - 8.36T + 73T^{2} \)
79 \( 1 + 15.1T + 79T^{2} \)
83 \( 1 - 3.55T + 83T^{2} \)
89 \( 1 - 0.732T + 89T^{2} \)
97 \( 1 - 1.75T + 97T^{2} \)
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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

−7.58609194973235282421177317707, −6.53928602821090878973529778583, −6.44389270812691826100822377372, −5.79938806561774021213635050176, −5.03352270953383442003103383735, −4.21504638443584018838846873101, −3.54747278973172317546332635735, −2.77041315277758713529678592268, −1.32370795982289362087531172755, −0.51642213846530661334004078393, 0.51642213846530661334004078393, 1.32370795982289362087531172755, 2.77041315277758713529678592268, 3.54747278973172317546332635735, 4.21504638443584018838846873101, 5.03352270953383442003103383735, 5.79938806561774021213635050176, 6.44389270812691826100822377372, 6.53928602821090878973529778583, 7.58609194973235282421177317707

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