Properties

Label 2-9680-1.1-c1-0-85
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  − 1.71·3-s − 5-s − 3.70·7-s − 0.0663·9-s − 1.17·13-s + 1.71·15-s − 5.40·17-s + 3.84·19-s + 6.34·21-s + 8.68·23-s + 25-s + 5.25·27-s − 6.87·29-s − 8.75·31-s + 3.70·35-s + 2.25·37-s + 2.01·39-s + 1.59·41-s − 4.11·43-s + 0.0663·45-s + 12.6·47-s + 6.72·49-s + 9.24·51-s − 12.3·53-s − 6.59·57-s + 0.393·59-s + 1.80·61-s + ⋯
L(s)  = 1  − 0.988·3-s − 0.447·5-s − 1.40·7-s − 0.0221·9-s − 0.326·13-s + 0.442·15-s − 1.30·17-s + 0.882·19-s + 1.38·21-s + 1.81·23-s + 0.200·25-s + 1.01·27-s − 1.27·29-s − 1.57·31-s + 0.626·35-s + 0.370·37-s + 0.322·39-s + 0.249·41-s − 0.628·43-s + 0.00988·45-s + 1.84·47-s + 0.961·49-s + 1.29·51-s − 1.70·53-s − 0.872·57-s + 0.0512·59-s + 0.230·61-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 \( 1 + 1.71T + 3T^{2} \)
7 \( 1 + 3.70T + 7T^{2} \)
13 \( 1 + 1.17T + 13T^{2} \)
17 \( 1 + 5.40T + 17T^{2} \)
19 \( 1 - 3.84T + 19T^{2} \)
23 \( 1 - 8.68T + 23T^{2} \)
29 \( 1 + 6.87T + 29T^{2} \)
31 \( 1 + 8.75T + 31T^{2} \)
37 \( 1 - 2.25T + 37T^{2} \)
41 \( 1 - 1.59T + 41T^{2} \)
43 \( 1 + 4.11T + 43T^{2} \)
47 \( 1 - 12.6T + 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 - 0.393T + 59T^{2} \)
61 \( 1 - 1.80T + 61T^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 - 6.85T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 - 1.11T + 79T^{2} \)
83 \( 1 + 0.0173T + 83T^{2} \)
89 \( 1 + 8.49T + 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−7.00813067157943336744544987742, −6.79737535129334186146118225064, −5.96005481883849285713125578146, −5.34658067208332929402750963912, −4.73546110619388920337107468216, −3.72536387819884001846918972116, −3.16811368285944879949088050837, −2.25308125148761492213903657386, −0.798752081427499377572076430348, 0, 0.798752081427499377572076430348, 2.25308125148761492213903657386, 3.16811368285944879949088050837, 3.72536387819884001846918972116, 4.73546110619388920337107468216, 5.34658067208332929402750963912, 5.96005481883849285713125578146, 6.79737535129334186146118225064, 7.00813067157943336744544987742

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