Properties

Label 2-7728-1.1-c1-0-12
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 − 3.61·5-s − 7-s + 9-s + 11-s + 5.09·13-s + 3.61·15-s − 5·17-s + 3.47·19-s + 21-s − 23-s + 8.09·25-s − 27-s − 6.23·29-s + 8.70·31-s − 33-s + 3.61·35-s + 1.47·37-s − 5.09·39-s − 5.76·41-s − 6.32·43-s − 3.61·45-s − 3.70·47-s + 49-s + 5·51-s + 0.381·53-s − 3.61·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.61·5-s − 0.377·7-s + 0.333·9-s + 0.301·11-s + 1.41·13-s + 0.934·15-s − 1.21·17-s + 0.796·19-s + 0.218·21-s − 0.208·23-s + 1.61·25-s − 0.192·27-s − 1.15·29-s + 1.56·31-s − 0.174·33-s + 0.611·35-s + 0.242·37-s − 0.815·39-s − 0.900·41-s − 0.964·43-s − 0.539·45-s − 0.540·47-s + 0.142·49-s + 0.700·51-s + 0.0524·53-s − 0.487·55-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

\(L(1)\) \(\approx\) \(0.7978637677\)
\(L(\frac12)\) \(\approx\) \(0.7978637677\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.61T + 5T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 - 5.09T + 13T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 - 3.47T + 19T^{2} \)
29 \( 1 + 6.23T + 29T^{2} \)
31 \( 1 - 8.70T + 31T^{2} \)
37 \( 1 - 1.47T + 37T^{2} \)
41 \( 1 + 5.76T + 41T^{2} \)
43 \( 1 + 6.32T + 43T^{2} \)
47 \( 1 + 3.70T + 47T^{2} \)
53 \( 1 - 0.381T + 53T^{2} \)
59 \( 1 - 3.61T + 59T^{2} \)
61 \( 1 + 7.56T + 61T^{2} \)
67 \( 1 - 7.32T + 67T^{2} \)
71 \( 1 - 4.32T + 71T^{2} \)
73 \( 1 + 0.527T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 9.18T + 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−8.015672636755053911095907203718, −6.95115601489987768791253286869, −6.69017447137998912765507837661, −5.84920794602332160283740966383, −4.94989970894876461359380500093, −4.16797382316433369939937273686, −3.72111112863357672990138547861, −2.94830031917551792198713365863, −1.52090931534457295814492378661, −0.47940065137436031845109445852, 0.47940065137436031845109445852, 1.52090931534457295814492378661, 2.94830031917551792198713365863, 3.72111112863357672990138547861, 4.16797382316433369939937273686, 4.94989970894876461359380500093, 5.84920794602332160283740966383, 6.69017447137998912765507837661, 6.95115601489987768791253286869, 8.015672636755053911095907203718

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