Properties

Label 2-9680-1.1-c1-0-159
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  + 1.60·3-s − 5-s − 4.33·7-s − 0.439·9-s − 0.439·13-s − 1.60·15-s + 7.10·19-s − 6.93·21-s + 3.90·23-s + 25-s − 5.50·27-s + 4·29-s − 8.22·31-s + 4.33·35-s + 0.878·37-s − 0.703·39-s + 10.8·41-s + 10.4·43-s + 0.439·45-s + 0.478·47-s + 11.7·49-s − 1.63·53-s + 11.3·57-s + 2.70·59-s − 12.0·61-s + 1.90·63-s + 0.439·65-s + ⋯
L(s)  = 1  + 0.923·3-s − 0.447·5-s − 1.63·7-s − 0.146·9-s − 0.121·13-s − 0.413·15-s + 1.62·19-s − 1.51·21-s + 0.813·23-s + 0.200·25-s − 1.05·27-s + 0.742·29-s − 1.47·31-s + 0.732·35-s + 0.144·37-s − 0.112·39-s + 1.69·41-s + 1.58·43-s + 0.0654·45-s + 0.0698·47-s + 1.68·49-s − 0.225·53-s + 1.50·57-s + 0.351·59-s − 1.54·61-s + 0.239·63-s + 0.0544·65-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 \( 1 - 1.60T + 3T^{2} \)
7 \( 1 + 4.33T + 7T^{2} \)
13 \( 1 + 0.439T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 7.10T + 19T^{2} \)
23 \( 1 - 3.90T + 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 8.22T + 31T^{2} \)
37 \( 1 - 0.878T + 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 - 0.478T + 47T^{2} \)
53 \( 1 + 1.63T + 53T^{2} \)
59 \( 1 - 2.70T + 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 + 5.67T + 67T^{2} \)
71 \( 1 + 16.0T + 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 - 3.78T + 79T^{2} \)
83 \( 1 + 3.50T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−7.44074032205800343627946764307, −6.84707313942262422567363367699, −5.94077380068457339052436533705, −5.44359917541206261409005861065, −4.28985920084398971814141238867, −3.61861444713474274083243338314, −2.92965251601321564998494036210, −2.65847384216052408046896124378, −1.16661977709532373427471207514, 0, 1.16661977709532373427471207514, 2.65847384216052408046896124378, 2.92965251601321564998494036210, 3.61861444713474274083243338314, 4.28985920084398971814141238867, 5.44359917541206261409005861065, 5.94077380068457339052436533705, 6.84707313942262422567363367699, 7.44074032205800343627946764307

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