Properties

Label 2-9680-1.1-c1-0-196
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  + 2.82·3-s − 5-s − 2·7-s + 5.00·9-s + 1.17·13-s − 2.82·15-s − 6.82·17-s − 5.65·21-s + 2.82·23-s + 25-s + 5.65·27-s + 3.65·29-s + 2·35-s − 7.65·37-s + 3.31·39-s − 6·41-s − 6·43-s − 5.00·45-s − 2.82·47-s − 3·49-s − 19.3·51-s + 11.6·53-s − 1.65·59-s + 9.31·61-s − 10.0·63-s − 1.17·65-s − 12.4·67-s + ⋯
L(s)  = 1  + 1.63·3-s − 0.447·5-s − 0.755·7-s + 1.66·9-s + 0.324·13-s − 0.730·15-s − 1.65·17-s − 1.23·21-s + 0.589·23-s + 0.200·25-s + 1.08·27-s + 0.679·29-s + 0.338·35-s − 1.25·37-s + 0.530·39-s − 0.937·41-s − 0.914·43-s − 0.745·45-s − 0.412·47-s − 0.428·49-s − 2.70·51-s + 1.60·53-s − 0.215·59-s + 1.19·61-s − 1.25·63-s − 0.145·65-s − 1.52·67-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: $\chi_{9680} (1, \cdot )$
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 \( 1 - 2.82T + 3T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
13 \( 1 - 1.17T + 13T^{2} \)
17 \( 1 + 6.82T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 7.65T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + 1.65T + 59T^{2} \)
61 \( 1 - 9.31T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 1.17T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 - 3.65T + 97T^{2} \)
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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

−7.27424107664339009375534040876, −6.92651807524227738868224020017, −6.24116275827303572947426936530, −5.06145366466741594609330374741, −4.31135154274888583666124147644, −3.62849220187349808216191674271, −3.06618025614208884707374095689, −2.38127189102190855728440602379, −1.48813123601387230128999333349, 0, 1.48813123601387230128999333349, 2.38127189102190855728440602379, 3.06618025614208884707374095689, 3.62849220187349808216191674271, 4.31135154274888583666124147644, 5.06145366466741594609330374741, 6.24116275827303572947426936530, 6.92651807524227738868224020017, 7.27424107664339009375534040876

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