Properties

Label 2-9680-1.1-c1-0-138
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  − 0.0947·3-s − 5-s + 0.826·7-s − 2.99·9-s − 2.99·13-s + 0.0947·15-s − 0.662·19-s − 0.0783·21-s − 0.473·23-s + 25-s + 0.567·27-s + 4·29-s + 4.64·31-s − 0.826·35-s + 5.98·37-s + 0.283·39-s − 0.394·41-s + 6.96·43-s + 2.99·45-s + 3.88·47-s − 6.31·49-s − 0.801·53-s + 0.0628·57-s + 1.71·59-s + 2.58·61-s − 2.47·63-s + 2.99·65-s + ⋯
L(s)  = 1  − 0.0547·3-s − 0.447·5-s + 0.312·7-s − 0.997·9-s − 0.829·13-s + 0.0244·15-s − 0.152·19-s − 0.0171·21-s − 0.0986·23-s + 0.200·25-s + 0.109·27-s + 0.742·29-s + 0.834·31-s − 0.139·35-s + 0.983·37-s + 0.0453·39-s − 0.0616·41-s + 1.06·43-s + 0.445·45-s + 0.567·47-s − 0.902·49-s − 0.110·53-s + 0.00831·57-s + 0.223·59-s + 0.330·61-s − 0.311·63-s + 0.370·65-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 \( 1 + 0.0947T + 3T^{2} \)
7 \( 1 - 0.826T + 7T^{2} \)
13 \( 1 + 2.99T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 0.662T + 19T^{2} \)
23 \( 1 + 0.473T + 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 - 4.64T + 31T^{2} \)
37 \( 1 - 5.98T + 37T^{2} \)
41 \( 1 + 0.394T + 41T^{2} \)
43 \( 1 - 6.96T + 43T^{2} \)
47 \( 1 - 3.88T + 47T^{2} \)
53 \( 1 + 0.801T + 53T^{2} \)
59 \( 1 - 1.71T + 59T^{2} \)
61 \( 1 - 2.58T + 61T^{2} \)
67 \( 1 + 5.69T + 67T^{2} \)
71 \( 1 - 5.59T + 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + 5.90T + 83T^{2} \)
89 \( 1 - 4.32T + 89T^{2} \)
97 \( 1 - 6.50T + 97T^{2} \)
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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

−7.44175805105501384523969919208, −6.63934058696791089135657305147, −5.99495504606142971205592784392, −5.22266987273651314427025890120, −4.60570777167381269508270313977, −3.88043908233057128780939454243, −2.85913156468548996137064698707, −2.40893930675955047539061388361, −1.08980022643092505653198845159, 0, 1.08980022643092505653198845159, 2.40893930675955047539061388361, 2.85913156468548996137064698707, 3.88043908233057128780939454243, 4.60570777167381269508270313977, 5.22266987273651314427025890120, 5.99495504606142971205592784392, 6.63934058696791089135657305147, 7.44175805105501384523969919208

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