Properties

Label 2-7728-1.1-c1-0-124
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 + 2.47·5-s − 7-s + 9-s − 5.95·11-s − 0.137·13-s + 2.47·15-s + 4.61·17-s − 19-s − 21-s − 23-s + 1.13·25-s + 27-s − 1.65·29-s + 2.61·31-s − 5.95·33-s − 2.47·35-s − 11.5·37-s − 0.137·39-s − 3.68·41-s − 11.0·43-s + 2.47·45-s − 5.34·47-s + 49-s + 4.61·51-s − 5.13·53-s − 14.7·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.10·5-s − 0.377·7-s + 0.333·9-s − 1.79·11-s − 0.0380·13-s + 0.639·15-s + 1.11·17-s − 0.229·19-s − 0.218·21-s − 0.208·23-s + 0.227·25-s + 0.192·27-s − 0.308·29-s + 0.469·31-s − 1.03·33-s − 0.418·35-s − 1.90·37-s − 0.0219·39-s − 0.574·41-s − 1.69·43-s + 0.369·45-s − 0.778·47-s + 0.142·49-s + 0.646·51-s − 0.705·53-s − 1.98·55-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: $\chi_{7728} (1, \cdot )$
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 \( 1 - 2.47T + 5T^{2} \)
11 \( 1 + 5.95T + 11T^{2} \)
13 \( 1 + 0.137T + 13T^{2} \)
17 \( 1 - 4.61T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
29 \( 1 + 1.65T + 29T^{2} \)
31 \( 1 - 2.61T + 31T^{2} \)
37 \( 1 + 11.5T + 37T^{2} \)
41 \( 1 + 3.68T + 41T^{2} \)
43 \( 1 + 11.0T + 43T^{2} \)
47 \( 1 + 5.34T + 47T^{2} \)
53 \( 1 + 5.13T + 53T^{2} \)
59 \( 1 + 5.13T + 59T^{2} \)
61 \( 1 - 4.52T + 61T^{2} \)
67 \( 1 + 2.47T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 0.340T + 73T^{2} \)
79 \( 1 - 12.5T + 79T^{2} \)
83 \( 1 + 1.65T + 83T^{2} \)
89 \( 1 - 3.86T + 89T^{2} \)
97 \( 1 + 2.88T + 97T^{2} \)
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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

−7.67179484117481195596942669088, −6.79403890883695540593021040575, −6.14166999140003955803396496304, −5.19207350813321755905914329741, −5.09014532184322965780287452511, −3.65471431556590167478210916768, −3.04618585280152059349048963970, −2.26883984666487826042679093015, −1.54729441987632912319477329718, 0, 1.54729441987632912319477329718, 2.26883984666487826042679093015, 3.04618585280152059349048963970, 3.65471431556590167478210916768, 5.09014532184322965780287452511, 5.19207350813321755905914329741, 6.14166999140003955803396496304, 6.79403890883695540593021040575, 7.67179484117481195596942669088

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