Properties

Label 2-9680-1.1-c1-0-197
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.29·3-s − 5-s − 3.51·7-s + 7.84·9-s − 4.27·13-s − 3.29·15-s + 1.58·17-s − 5.55·19-s − 11.5·21-s + 6.39·23-s + 25-s + 15.9·27-s − 0.381·29-s + 0.760·31-s + 3.51·35-s − 1.45·37-s − 14.0·39-s − 6.12·41-s + 7.19·43-s − 7.84·45-s − 4.32·47-s + 5.35·49-s + 5.23·51-s − 6.94·53-s − 18.2·57-s − 12.7·59-s − 8.89·61-s + ⋯
L(s)  = 1  + 1.90·3-s − 0.447·5-s − 1.32·7-s + 2.61·9-s − 1.18·13-s − 0.850·15-s + 0.385·17-s − 1.27·19-s − 2.52·21-s + 1.33·23-s + 0.200·25-s + 3.06·27-s − 0.0707·29-s + 0.136·31-s + 0.594·35-s − 0.238·37-s − 2.25·39-s − 0.956·41-s + 1.09·43-s − 1.16·45-s − 0.630·47-s + 0.765·49-s + 0.733·51-s − 0.954·53-s − 2.42·57-s − 1.65·59-s − 1.13·61-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 \( 1 - 3.29T + 3T^{2} \)
7 \( 1 + 3.51T + 7T^{2} \)
13 \( 1 + 4.27T + 13T^{2} \)
17 \( 1 - 1.58T + 17T^{2} \)
19 \( 1 + 5.55T + 19T^{2} \)
23 \( 1 - 6.39T + 23T^{2} \)
29 \( 1 + 0.381T + 29T^{2} \)
31 \( 1 - 0.760T + 31T^{2} \)
37 \( 1 + 1.45T + 37T^{2} \)
41 \( 1 + 6.12T + 41T^{2} \)
43 \( 1 - 7.19T + 43T^{2} \)
47 \( 1 + 4.32T + 47T^{2} \)
53 \( 1 + 6.94T + 53T^{2} \)
59 \( 1 + 12.7T + 59T^{2} \)
61 \( 1 + 8.89T + 61T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 + 7.35T + 71T^{2} \)
73 \( 1 - 7.03T + 73T^{2} \)
79 \( 1 - 6.99T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 - 0.451T + 89T^{2} \)
97 \( 1 + 2.77T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.56407574011610342054569043447, −6.81357914386344755822677122334, −6.34275755998286011389602784626, −4.97522395772521669344784672390, −4.35198010783739888961178684447, −3.54760376391707084575468605902, −3.00204174733662304956506749221, −2.52226713703555329241337771878, −1.49024134485997359212068767904, 0, 1.49024134485997359212068767904, 2.52226713703555329241337771878, 3.00204174733662304956506749221, 3.54760376391707084575468605902, 4.35198010783739888961178684447, 4.97522395772521669344784672390, 6.34275755998286011389602784626, 6.81357914386344755822677122334, 7.56407574011610342054569043447

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