Properties

Label 2-9680-1.1-c1-0-147
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.41·3-s + 5-s + 2.41·7-s + 2.82·9-s − 2·13-s − 2.41·15-s − 5.65·17-s + 4.82·19-s − 5.82·21-s − 3.65·23-s + 25-s + 0.414·27-s − 2·29-s + 5.65·31-s + 2.41·35-s − 4·37-s + 4.82·39-s + 9.48·41-s + 3.58·43-s + 2.82·45-s − 7.58·47-s − 1.17·49-s + 13.6·51-s − 7.65·53-s − 11.6·57-s + 11.3·59-s − 61-s + ⋯
L(s)  = 1  − 1.39·3-s + 0.447·5-s + 0.912·7-s + 0.942·9-s − 0.554·13-s − 0.623·15-s − 1.37·17-s + 1.10·19-s − 1.27·21-s − 0.762·23-s + 0.200·25-s + 0.0797·27-s − 0.371·29-s + 1.01·31-s + 0.408·35-s − 0.657·37-s + 0.773·39-s + 1.48·41-s + 0.546·43-s + 0.421·45-s − 1.10·47-s − 0.167·49-s + 1.91·51-s − 1.05·53-s − 1.54·57-s + 1.47·59-s − 0.128·61-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: \(1\)
Selberg data: \((2,\ 9680,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.41T + 3T^{2} \)
7 \( 1 - 2.41T + 7T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 5.65T + 17T^{2} \)
19 \( 1 - 4.82T + 19T^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 - 9.48T + 41T^{2} \)
43 \( 1 - 3.58T + 43T^{2} \)
47 \( 1 + 7.58T + 47T^{2} \)
53 \( 1 + 7.65T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 - 6.41T + 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 - 2.65T + 89T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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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

−7.10050352974757718623387664199, −6.64108838898045297503782033631, −5.80558353300617983734879709967, −5.40064632878912385465737015058, −4.66500485957538367594639257405, −4.23354305160660310076416562911, −2.89146151581615367550715677658, −1.99058359999203636694300086501, −1.11699565031101077599152916895, 0, 1.11699565031101077599152916895, 1.99058359999203636694300086501, 2.89146151581615367550715677658, 4.23354305160660310076416562911, 4.66500485957538367594639257405, 5.40064632878912385465737015058, 5.80558353300617983734879709967, 6.64108838898045297503782033631, 7.10050352974757718623387664199

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