Properties

Label 2-7728-1.1-c1-0-49
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 + 2.61·5-s + 7-s + 9-s + 3.43·11-s + 0.821·13-s − 2.61·15-s − 5.79·17-s + 7.43·19-s − 21-s − 23-s + 1.82·25-s − 27-s − 5.79·29-s + 3.07·31-s − 3.43·33-s + 2.61·35-s − 0.922·37-s − 0.821·39-s + 0.566·41-s + 11.6·43-s + 2.61·45-s + 2.35·47-s + 49-s + 5.79·51-s + 0.968·53-s + 8.96·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.16·5-s + 0.377·7-s + 0.333·9-s + 1.03·11-s + 0.227·13-s − 0.674·15-s − 1.40·17-s + 1.70·19-s − 0.218·21-s − 0.208·23-s + 0.364·25-s − 0.192·27-s − 1.07·29-s + 0.552·31-s − 0.597·33-s + 0.441·35-s − 0.151·37-s − 0.131·39-s + 0.0884·41-s + 1.78·43-s + 0.389·45-s + 0.343·47-s + 0.142·49-s + 0.810·51-s + 0.132·53-s + 1.20·55-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

\(L(1)\) \(\approx\) \(2.541082165\)
\(L(\frac12)\) \(\approx\) \(2.541082165\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.61T + 5T^{2} \)
11 \( 1 - 3.43T + 11T^{2} \)
13 \( 1 - 0.821T + 13T^{2} \)
17 \( 1 + 5.79T + 17T^{2} \)
19 \( 1 - 7.43T + 19T^{2} \)
29 \( 1 + 5.79T + 29T^{2} \)
31 \( 1 - 3.07T + 31T^{2} \)
37 \( 1 + 0.922T + 37T^{2} \)
41 \( 1 - 0.566T + 41T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 - 2.35T + 47T^{2} \)
53 \( 1 - 0.968T + 53T^{2} \)
59 \( 1 + 1.89T + 59T^{2} \)
61 \( 1 - 2.40T + 61T^{2} \)
67 \( 1 - 4.10T + 67T^{2} \)
71 \( 1 + 1.17T + 71T^{2} \)
73 \( 1 - 8.14T + 73T^{2} \)
79 \( 1 - 2.56T + 79T^{2} \)
83 \( 1 - 6.30T + 83T^{2} \)
89 \( 1 + 7.89T + 89T^{2} \)
97 \( 1 + 1.79T + 97T^{2} \)
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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

−7.71340925380666275522393481975, −7.04546963904112918211790020336, −6.35737638883119377242503889056, −5.80975771277602586128946132360, −5.21630668360250801978224669582, −4.39549268481482109913874999577, −3.65047376048885859974614144152, −2.46432325897463466208635351480, −1.71158656745884287684929701431, −0.867859080206768168506541000624, 0.867859080206768168506541000624, 1.71158656745884287684929701431, 2.46432325897463466208635351480, 3.65047376048885859974614144152, 4.39549268481482109913874999577, 5.21630668360250801978224669582, 5.80975771277602586128946132360, 6.35737638883119377242503889056, 7.04546963904112918211790020336, 7.71340925380666275522393481975

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