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Properties

Label 2-9680-1.1-c1-0-157
Degree $2$
Conductor $9680$
Sign $-1$
Analytic cond. $77.2951$
Root an. cond. $8.79176$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 5-s + 7-s + 6·9-s + 6·13-s − 3·15-s − 3·17-s − 5·19-s − 3·21-s + 2·23-s + 25-s − 9·27-s + 5·29-s − 5·31-s + 35-s − 37-s − 18·39-s + 2·41-s + 12·43-s + 6·45-s + 2·47-s − 6·49-s + 9·51-s − 13·53-s + 15·57-s − 2·59-s − 61-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s + 1.66·13-s − 0.774·15-s − 0.727·17-s − 1.14·19-s − 0.654·21-s + 0.417·23-s + 1/5·25-s − 1.73·27-s + 0.928·29-s − 0.898·31-s + 0.169·35-s − 0.164·37-s − 2.88·39-s + 0.312·41-s + 1.82·43-s + 0.894·45-s + 0.291·47-s − 6/7·49-s + 1.26·51-s − 1.78·53-s + 1.98·57-s − 0.260·59-s − 0.128·61-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: yes
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 \( 1 + p T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−7.05489758355601967759943436966, −6.40685004287654883435103180679, −6.00514674241637878671719902267, −5.50897498234508480061637227522, −4.43234051750562207876488859604, −4.35957910011061397234039659849, −3.04994318437691480144006708745, −1.78022016543760968230938159437, −1.15905119202116105706264313535, 0, 1.15905119202116105706264313535, 1.78022016543760968230938159437, 3.04994318437691480144006708745, 4.35957910011061397234039659849, 4.43234051750562207876488859604, 5.50897498234508480061637227522, 6.00514674241637878671719902267, 6.40685004287654883435103180679, 7.05489758355601967759943436966

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