Properties

Label 2-9680-1.1-c1-0-53
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  + 0.873·3-s + 5-s − 3.64·7-s − 2.23·9-s + 1.06·13-s + 0.873·15-s − 0.252·17-s + 3.06·19-s − 3.18·21-s + 7.04·23-s + 25-s − 4.57·27-s − 1.61·29-s + 0.788·31-s − 3.64·35-s − 4.91·37-s + 0.932·39-s − 6.39·41-s − 2.63·43-s − 2.23·45-s + 7.78·47-s + 6.25·49-s − 0.220·51-s − 6.32·53-s + 2.67·57-s + 5.48·59-s + 6.80·61-s + ⋯
L(s)  = 1  + 0.504·3-s + 0.447·5-s − 1.37·7-s − 0.745·9-s + 0.296·13-s + 0.225·15-s − 0.0613·17-s + 0.703·19-s − 0.693·21-s + 1.46·23-s + 0.200·25-s − 0.880·27-s − 0.299·29-s + 0.141·31-s − 0.615·35-s − 0.807·37-s + 0.149·39-s − 0.998·41-s − 0.401·43-s − 0.333·45-s + 1.13·47-s + 0.893·49-s − 0.0309·51-s − 0.868·53-s + 0.354·57-s + 0.713·59-s + 0.870·61-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

\(L(1)\) \(\approx\) \(1.869860154\)
\(L(\frac12)\) \(\approx\) \(1.869860154\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.873T + 3T^{2} \)
7 \( 1 + 3.64T + 7T^{2} \)
13 \( 1 - 1.06T + 13T^{2} \)
17 \( 1 + 0.252T + 17T^{2} \)
19 \( 1 - 3.06T + 19T^{2} \)
23 \( 1 - 7.04T + 23T^{2} \)
29 \( 1 + 1.61T + 29T^{2} \)
31 \( 1 - 0.788T + 31T^{2} \)
37 \( 1 + 4.91T + 37T^{2} \)
41 \( 1 + 6.39T + 41T^{2} \)
43 \( 1 + 2.63T + 43T^{2} \)
47 \( 1 - 7.78T + 47T^{2} \)
53 \( 1 + 6.32T + 53T^{2} \)
59 \( 1 - 5.48T + 59T^{2} \)
61 \( 1 - 6.80T + 61T^{2} \)
67 \( 1 - 9.49T + 67T^{2} \)
71 \( 1 + 7.53T + 71T^{2} \)
73 \( 1 + 16.4T + 73T^{2} \)
79 \( 1 + 3.02T + 79T^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 - 8.77T + 97T^{2} \)
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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

−7.64052507375680790685281054936, −6.89135784881070959356785484973, −6.43178330653290069350163573597, −5.60214905064840951604016556732, −5.13990414817422510673408947131, −3.95487242877757639213134546565, −3.16926007544645891191397091947, −2.89845858891334464783038859575, −1.83927191219932171886193159918, −0.61609121270106764248685991596, 0.61609121270106764248685991596, 1.83927191219932171886193159918, 2.89845858891334464783038859575, 3.16926007544645891191397091947, 3.95487242877757639213134546565, 5.13990414817422510673408947131, 5.60214905064840951604016556732, 6.43178330653290069350163573597, 6.89135784881070959356785484973, 7.64052507375680790685281054936

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