Properties

Label 2-7728-1.1-c1-0-104
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 0.381·5-s − 7-s + 9-s − 2.23·11-s − 0.145·13-s − 0.381·15-s − 5.47·17-s + 8.23·19-s − 21-s − 23-s − 4.85·25-s + 27-s − 2.70·29-s + 5·31-s − 2.23·33-s + 0.381·35-s − 0.527·37-s − 0.145·39-s + 8.70·41-s + 8.32·43-s − 0.381·45-s − 5.23·47-s + 49-s − 5.47·51-s − 3.61·53-s + 0.854·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.170·5-s − 0.377·7-s + 0.333·9-s − 0.674·11-s − 0.0404·13-s − 0.0986·15-s − 1.32·17-s + 1.88·19-s − 0.218·21-s − 0.208·23-s − 0.970·25-s + 0.192·27-s − 0.502·29-s + 0.898·31-s − 0.389·33-s + 0.0645·35-s − 0.0867·37-s − 0.0233·39-s + 1.35·41-s + 1.26·43-s − 0.0569·45-s − 0.763·47-s + 0.142·49-s − 0.766·51-s − 0.496·53-s + 0.115·55-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: \(1\)
Selberg data: \((2,\ 7728,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
7 \( 1 + T \)
23 \( 1 + T \)
good5 \( 1 + 0.381T + 5T^{2} \)
11 \( 1 + 2.23T + 11T^{2} \)
13 \( 1 + 0.145T + 13T^{2} \)
17 \( 1 + 5.47T + 17T^{2} \)
19 \( 1 - 8.23T + 19T^{2} \)
29 \( 1 + 2.70T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + 0.527T + 37T^{2} \)
41 \( 1 - 8.70T + 41T^{2} \)
43 \( 1 - 8.32T + 43T^{2} \)
47 \( 1 + 5.23T + 47T^{2} \)
53 \( 1 + 3.61T + 53T^{2} \)
59 \( 1 - 4.85T + 59T^{2} \)
61 \( 1 + 9.09T + 61T^{2} \)
67 \( 1 - 6.85T + 67T^{2} \)
71 \( 1 + 9.38T + 71T^{2} \)
73 \( 1 + 3.47T + 73T^{2} \)
79 \( 1 + 5.94T + 79T^{2} \)
83 \( 1 + 7.94T + 83T^{2} \)
89 \( 1 + 16.3T + 89T^{2} \)
97 \( 1 + 17.1T + 97T^{2} \)
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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

−7.52412292762567520064767913132, −7.04755057467137186395322827729, −6.08917510380939203564963365080, −5.47565733543702613294703081691, −4.53851231323841241013272825803, −3.92451885214682736216044604619, −2.97386399381430501815190676565, −2.46745426239782070216495055304, −1.32580720577385318735724055540, 0, 1.32580720577385318735724055540, 2.46745426239782070216495055304, 2.97386399381430501815190676565, 3.92451885214682736216044604619, 4.53851231323841241013272825803, 5.47565733543702613294703081691, 6.08917510380939203564963365080, 7.04755057467137186395322827729, 7.52412292762567520064767913132

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