Properties

Label 2-9680-1.1-c1-0-31
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  − 2.19·3-s + 5-s + 2.58·7-s + 1.83·9-s − 1.66·13-s − 2.19·15-s − 6.39·17-s + 0.330·19-s − 5.68·21-s − 1.33·23-s + 25-s + 2.56·27-s − 0.941·29-s − 5.78·31-s + 2.58·35-s − 7.41·37-s + 3.66·39-s − 5.49·41-s − 4.54·43-s + 1.83·45-s + 7.21·47-s − 0.318·49-s + 14.0·51-s + 6.55·53-s − 0.727·57-s + 7.74·59-s − 3.88·61-s + ⋯
L(s)  = 1  − 1.26·3-s + 0.447·5-s + 0.976·7-s + 0.610·9-s − 0.462·13-s − 0.567·15-s − 1.55·17-s + 0.0759·19-s − 1.23·21-s − 0.278·23-s + 0.200·25-s + 0.493·27-s − 0.174·29-s − 1.03·31-s + 0.436·35-s − 1.21·37-s + 0.587·39-s − 0.857·41-s − 0.692·43-s + 0.273·45-s + 1.05·47-s − 0.0455·49-s + 1.96·51-s + 0.900·53-s − 0.0963·57-s + 1.00·59-s − 0.497·61-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

\(L(1)\) \(\approx\) \(1.028157490\)
\(L(\frac12)\) \(\approx\) \(1.028157490\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.19T + 3T^{2} \)
7 \( 1 - 2.58T + 7T^{2} \)
13 \( 1 + 1.66T + 13T^{2} \)
17 \( 1 + 6.39T + 17T^{2} \)
19 \( 1 - 0.330T + 19T^{2} \)
23 \( 1 + 1.33T + 23T^{2} \)
29 \( 1 + 0.941T + 29T^{2} \)
31 \( 1 + 5.78T + 31T^{2} \)
37 \( 1 + 7.41T + 37T^{2} \)
41 \( 1 + 5.49T + 41T^{2} \)
43 \( 1 + 4.54T + 43T^{2} \)
47 \( 1 - 7.21T + 47T^{2} \)
53 \( 1 - 6.55T + 53T^{2} \)
59 \( 1 - 7.74T + 59T^{2} \)
61 \( 1 + 3.88T + 61T^{2} \)
67 \( 1 + 9.68T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 - 7.61T + 79T^{2} \)
83 \( 1 - 9.29T + 83T^{2} \)
89 \( 1 + 17.7T + 89T^{2} \)
97 \( 1 - 4.03T + 97T^{2} \)
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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

−7.45737484875808461095594385414, −6.88788174253785649059307324098, −6.28982898963978736252511568907, −5.50761448006506643540331230695, −5.05056825496073756465363661055, −4.52299549547613386036705173900, −3.57535606629843752895594177453, −2.29064083077210863846606483875, −1.72660905145800976341035456225, −0.50966081490343639699632997232, 0.50966081490343639699632997232, 1.72660905145800976341035456225, 2.29064083077210863846606483875, 3.57535606629843752895594177453, 4.52299549547613386036705173900, 5.05056825496073756465363661055, 5.50761448006506643540331230695, 6.28982898963978736252511568907, 6.88788174253785649059307324098, 7.45737484875808461095594385414

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