Properties

Label 2-7728-1.1-c1-0-24
Degree $2$
Conductor $7728$
Sign $1$
Analytic cond. $61.7083$
Root an. cond. $7.85546$
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  − 3-s + 1.38·5-s − 7-s + 9-s − 11-s − 1.61·13-s − 1.38·15-s + 3.47·17-s + 0.236·19-s + 21-s − 23-s − 3.09·25-s − 27-s + 6.23·29-s − 4.70·31-s + 33-s − 1.38·35-s − 4.23·37-s + 1.61·39-s + 5.47·41-s − 2.85·43-s + 1.38·45-s + 1.70·47-s + 49-s − 3.47·51-s + 11.0·53-s − 1.38·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.618·5-s − 0.377·7-s + 0.333·9-s − 0.301·11-s − 0.448·13-s − 0.356·15-s + 0.842·17-s + 0.0541·19-s + 0.218·21-s − 0.208·23-s − 0.618·25-s − 0.192·27-s + 1.15·29-s − 0.845·31-s + 0.174·33-s − 0.233·35-s − 0.696·37-s + 0.259·39-s + 0.854·41-s − 0.435·43-s + 0.206·45-s + 0.249·47-s + 0.142·49-s − 0.486·51-s + 1.52·53-s − 0.186·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7728\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(61.7083\)
Root analytic conductor: \(7.85546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7728,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.518111301\)
\(L(\frac12)\) \(\approx\) \(1.518111301\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 + T \)
good5 \( 1 - 1.38T + 5T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + 1.61T + 13T^{2} \)
17 \( 1 - 3.47T + 17T^{2} \)
19 \( 1 - 0.236T + 19T^{2} \)
29 \( 1 - 6.23T + 29T^{2} \)
31 \( 1 + 4.70T + 31T^{2} \)
37 \( 1 + 4.23T + 37T^{2} \)
41 \( 1 - 5.47T + 41T^{2} \)
43 \( 1 + 2.85T + 43T^{2} \)
47 \( 1 - 1.70T + 47T^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 - 1.38T + 59T^{2} \)
61 \( 1 + 1.14T + 61T^{2} \)
67 \( 1 + 3.09T + 67T^{2} \)
71 \( 1 + 5.32T + 71T^{2} \)
73 \( 1 - 6.23T + 73T^{2} \)
79 \( 1 - 9.76T + 79T^{2} \)
83 \( 1 - 0.0557T + 83T^{2} \)
89 \( 1 + 6.56T + 89T^{2} \)
97 \( 1 + 6.70T + 97T^{2} \)
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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

−7.72787803486407930426555838558, −7.12453849022559754265020899962, −6.39948085090525015781424389118, −5.69463065186954083543147569045, −5.28688048652044623719131175498, −4.39382637278833927012552564190, −3.53314053060907816163814217790, −2.62750113059897514735998441667, −1.75629663448124327733765620968, −0.63104045278478870217603426565, 0.63104045278478870217603426565, 1.75629663448124327733765620968, 2.62750113059897514735998441667, 3.53314053060907816163814217790, 4.39382637278833927012552564190, 5.28688048652044623719131175498, 5.69463065186954083543147569045, 6.39948085090525015781424389118, 7.12453849022559754265020899962, 7.72787803486407930426555838558

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