Properties

Label 2-9680-1.1-c1-0-100
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  + 2.29·3-s − 5-s + 4.66·7-s + 2.24·9-s − 5.94·13-s − 2.29·15-s + 3.30·17-s − 4.19·19-s + 10.6·21-s + 1.71·23-s + 25-s − 1.72·27-s + 7.76·29-s + 2.97·31-s − 4.66·35-s − 5.68·37-s − 13.6·39-s + 6.14·41-s − 4.42·43-s − 2.24·45-s − 4.13·47-s + 14.7·49-s + 7.56·51-s + 8.10·53-s − 9.60·57-s + 6.21·59-s + 1.52·61-s + ⋯
L(s)  = 1  + 1.32·3-s − 0.447·5-s + 1.76·7-s + 0.749·9-s − 1.64·13-s − 0.591·15-s + 0.801·17-s − 0.961·19-s + 2.33·21-s + 0.356·23-s + 0.200·25-s − 0.331·27-s + 1.44·29-s + 0.533·31-s − 0.788·35-s − 0.934·37-s − 2.17·39-s + 0.960·41-s − 0.674·43-s − 0.335·45-s − 0.602·47-s + 2.10·49-s + 1.05·51-s + 1.11·53-s − 1.27·57-s + 0.809·59-s + 0.195·61-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

\(L(1)\) \(\approx\) \(3.812220311\)
\(L(\frac12)\) \(\approx\) \(3.812220311\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.29T + 3T^{2} \)
7 \( 1 - 4.66T + 7T^{2} \)
13 \( 1 + 5.94T + 13T^{2} \)
17 \( 1 - 3.30T + 17T^{2} \)
19 \( 1 + 4.19T + 19T^{2} \)
23 \( 1 - 1.71T + 23T^{2} \)
29 \( 1 - 7.76T + 29T^{2} \)
31 \( 1 - 2.97T + 31T^{2} \)
37 \( 1 + 5.68T + 37T^{2} \)
41 \( 1 - 6.14T + 41T^{2} \)
43 \( 1 + 4.42T + 43T^{2} \)
47 \( 1 + 4.13T + 47T^{2} \)
53 \( 1 - 8.10T + 53T^{2} \)
59 \( 1 - 6.21T + 59T^{2} \)
61 \( 1 - 1.52T + 61T^{2} \)
67 \( 1 - 7.46T + 67T^{2} \)
71 \( 1 - 2.23T + 71T^{2} \)
73 \( 1 - 7.49T + 73T^{2} \)
79 \( 1 - 17.5T + 79T^{2} \)
83 \( 1 - 11.0T + 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + 9.53T + 97T^{2} \)
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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

−7.84725826703312805272098586433, −7.32974371974318174579306667384, −6.56769610030496370391620904283, −5.26703395991572167355988546513, −4.88078531445661572640507623543, −4.18782314803278624090226430981, −3.37299411041616940301722342467, −2.42418297801609243715419115596, −2.06530081174531134182777892453, −0.879181702258480190261339690645, 0.879181702258480190261339690645, 2.06530081174531134182777892453, 2.42418297801609243715419115596, 3.37299411041616940301722342467, 4.18782314803278624090226430981, 4.88078531445661572640507623543, 5.26703395991572167355988546513, 6.56769610030496370391620904283, 7.32974371974318174579306667384, 7.84725826703312805272098586433

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