Properties

Label 2-9680-1.1-c1-0-23
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  − 2.62·3-s + 5-s − 2.62·7-s + 3.89·9-s + 1.72·13-s − 2.62·15-s − 0.626·17-s − 8.14·19-s + 6.89·21-s + 5.52·23-s + 25-s − 2.35·27-s + 2.89·29-s + 6.89·31-s − 2.62·35-s − 0.896·37-s − 4.54·39-s + 5.25·41-s − 9.52·43-s + 3.89·45-s + 0.270·47-s − 0.103·49-s + 1.64·51-s − 0.896·53-s + 21.4·57-s − 7.04·59-s − 9.40·61-s + ⋯
L(s)  = 1  − 1.51·3-s + 0.447·5-s − 0.992·7-s + 1.29·9-s + 0.479·13-s − 0.678·15-s − 0.151·17-s − 1.86·19-s + 1.50·21-s + 1.15·23-s + 0.200·25-s − 0.453·27-s + 0.537·29-s + 1.23·31-s − 0.443·35-s − 0.147·37-s − 0.727·39-s + 0.820·41-s − 1.45·43-s + 0.580·45-s + 0.0394·47-s − 0.0147·49-s + 0.230·51-s − 0.123·53-s + 2.83·57-s − 0.917·59-s − 1.20·61-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

\(L(1)\) \(\approx\) \(0.7468691845\)
\(L(\frac12)\) \(\approx\) \(0.7468691845\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.62T + 3T^{2} \)
7 \( 1 + 2.62T + 7T^{2} \)
13 \( 1 - 1.72T + 13T^{2} \)
17 \( 1 + 0.626T + 17T^{2} \)
19 \( 1 + 8.14T + 19T^{2} \)
23 \( 1 - 5.52T + 23T^{2} \)
29 \( 1 - 2.89T + 29T^{2} \)
31 \( 1 - 6.89T + 31T^{2} \)
37 \( 1 + 0.896T + 37T^{2} \)
41 \( 1 - 5.25T + 41T^{2} \)
43 \( 1 + 9.52T + 43T^{2} \)
47 \( 1 - 0.270T + 47T^{2} \)
53 \( 1 + 0.896T + 53T^{2} \)
59 \( 1 + 7.04T + 59T^{2} \)
61 \( 1 + 9.40T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 - 9.60T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−7.38562113611114220856632303498, −6.62887424590716179400721677078, −6.24550499630155809537551097231, −5.95407448680879077986627440024, −4.84870048181756176888336655361, −4.55886187081697257717942202768, −3.45631293117335009361436578699, −2.60603100049963760950165201406, −1.46775641696807008546089622876, −0.46297039293501990590908250169, 0.46297039293501990590908250169, 1.46775641696807008546089622876, 2.60603100049963760950165201406, 3.45631293117335009361436578699, 4.55886187081697257717942202768, 4.84870048181756176888336655361, 5.95407448680879077986627440024, 6.24550499630155809537551097231, 6.62887424590716179400721677078, 7.38562113611114220856632303498

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