Properties

Label 2-7728-1.1-c1-0-98
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 + 3.34·5-s + 7-s + 9-s + 4.36·11-s + 1.50·13-s + 3.34·15-s + 5.56·17-s + 4.36·19-s + 21-s + 23-s + 6.17·25-s + 27-s − 5.73·29-s − 10.9·31-s + 4.36·33-s + 3.34·35-s − 0.336·37-s + 1.50·39-s + 7.82·41-s + 9.58·43-s + 3.34·45-s − 9.39·47-s + 49-s + 5.56·51-s + 1.31·53-s + 14.6·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.49·5-s + 0.377·7-s + 0.333·9-s + 1.31·11-s + 0.418·13-s + 0.863·15-s + 1.34·17-s + 1.00·19-s + 0.218·21-s + 0.208·23-s + 1.23·25-s + 0.192·27-s − 1.06·29-s − 1.96·31-s + 0.760·33-s + 0.565·35-s − 0.0552·37-s + 0.241·39-s + 1.22·41-s + 1.46·43-s + 0.498·45-s − 1.36·47-s + 0.142·49-s + 0.779·51-s + 0.180·53-s + 1.96·55-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: \(0\)
Selberg data: \((2,\ 7728,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.649421816\)
\(L(\frac12)\) \(\approx\) \(4.649421816\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.34T + 5T^{2} \)
11 \( 1 - 4.36T + 11T^{2} \)
13 \( 1 - 1.50T + 13T^{2} \)
17 \( 1 - 5.56T + 17T^{2} \)
19 \( 1 - 4.36T + 19T^{2} \)
29 \( 1 + 5.73T + 29T^{2} \)
31 \( 1 + 10.9T + 31T^{2} \)
37 \( 1 + 0.336T + 37T^{2} \)
41 \( 1 - 7.82T + 41T^{2} \)
43 \( 1 - 9.58T + 43T^{2} \)
47 \( 1 + 9.39T + 47T^{2} \)
53 \( 1 - 1.31T + 53T^{2} \)
59 \( 1 - 2.02T + 59T^{2} \)
61 \( 1 - 4.87T + 61T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 9.23T + 73T^{2} \)
79 \( 1 + 0.878T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 - 8.31T + 97T^{2} \)
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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

−7.72437675184167379026696744291, −7.29425622344778751377906921674, −6.40626156667196894408667203533, −5.65182165513007924935389404460, −5.36711562455926100694596811189, −4.14936802458201204248223841843, −3.50192685545371827038647299079, −2.63039496883098828518661489211, −1.58883348292688686936527117726, −1.26941385472485703981730464892, 1.26941385472485703981730464892, 1.58883348292688686936527117726, 2.63039496883098828518661489211, 3.50192685545371827038647299079, 4.14936802458201204248223841843, 5.36711562455926100694596811189, 5.65182165513007924935389404460, 6.40626156667196894408667203533, 7.29425622344778751377906921674, 7.72437675184167379026696744291

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