Properties

Label 2-67760-1.1-c1-0-11
Degree $2$
Conductor $67760$
Sign $1$
Analytic cond. $541.066$
Root an. cond. $23.2608$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 7-s − 2·9-s − 5·13-s + 15-s − 3·17-s + 2·19-s − 21-s + 6·23-s + 25-s + 5·27-s − 3·29-s + 4·31-s − 35-s + 2·37-s + 5·39-s + 12·41-s − 10·43-s + 2·45-s − 9·47-s + 49-s + 3·51-s + 12·53-s − 2·57-s − 8·61-s − 2·63-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s − 1.38·13-s + 0.258·15-s − 0.727·17-s + 0.458·19-s − 0.218·21-s + 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s + 0.718·31-s − 0.169·35-s + 0.328·37-s + 0.800·39-s + 1.87·41-s − 1.52·43-s + 0.298·45-s − 1.31·47-s + 1/7·49-s + 0.420·51-s + 1.64·53-s − 0.264·57-s − 1.02·61-s − 0.251·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(67760\)    =    \(2^{4} \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(541.066\)
Root analytic conductor: \(23.2608\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 67760,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9537107077\)
\(L(\frac12)\) \(\approx\) \(0.9537107077\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 \)
good3 \( 1 + T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−14.24693011878691, −13.69648856622150, −13.06775347659684, −12.63555847478761, −12.02329012145661, −11.62038494371479, −11.27757819489321, −10.78376062730788, −10.21377841462648, −9.569241963943306, −9.098328284967463, −8.530488289028281, −7.951375453916034, −7.448388462856904, −6.888293816171589, −6.431868907244784, −5.647209694957754, −5.166103078219440, −4.719899708184433, −4.210569065855972, −3.276943354165636, −2.749556234913048, −2.153525597603477, −1.135932407218472, −0.3726268198924754, 0.3726268198924754, 1.135932407218472, 2.153525597603477, 2.749556234913048, 3.276943354165636, 4.210569065855972, 4.719899708184433, 5.166103078219440, 5.647209694957754, 6.431868907244784, 6.888293816171589, 7.448388462856904, 7.951375453916034, 8.530488289028281, 9.098328284967463, 9.569241963943306, 10.21377841462648, 10.78376062730788, 11.27757819489321, 11.62038494371479, 12.02329012145661, 12.63555847478761, 13.06775347659684, 13.69648856622150, 14.24693011878691

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