Properties

Label 2-7728-1.1-c1-0-97
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 + 2·5-s − 7-s + 9-s − 4.47·13-s − 2·15-s + 4.47·17-s − 2.47·19-s + 21-s + 23-s − 25-s − 27-s − 2·29-s + 2.47·31-s − 2·35-s + 6.94·37-s + 4.47·39-s − 6·41-s + 4.94·43-s + 2·45-s + 2.47·47-s + 49-s − 4.47·51-s − 10.9·53-s + 2.47·57-s − 4·59-s − 6·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.377·7-s + 0.333·9-s − 1.24·13-s − 0.516·15-s + 1.08·17-s − 0.567·19-s + 0.218·21-s + 0.208·23-s − 0.200·25-s − 0.192·27-s − 0.371·29-s + 0.444·31-s − 0.338·35-s + 1.14·37-s + 0.716·39-s − 0.937·41-s + 0.753·43-s + 0.298·45-s + 0.360·47-s + 0.142·49-s − 0.626·51-s − 1.50·53-s + 0.327·57-s − 0.520·59-s − 0.768·61-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 \( 1 - 2T + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 - 4.47T + 17T^{2} \)
19 \( 1 + 2.47T + 19T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 2.47T + 31T^{2} \)
37 \( 1 - 6.94T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4.94T + 43T^{2} \)
47 \( 1 - 2.47T + 47T^{2} \)
53 \( 1 + 10.9T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + 4.94T + 67T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 - 4.94T + 79T^{2} \)
83 \( 1 + 2.47T + 83T^{2} \)
89 \( 1 - 9.41T + 89T^{2} \)
97 \( 1 + 0.472T + 97T^{2} \)
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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

−7.57217738192382147009034730617, −6.59887555103524585392383492632, −6.17820793698472085055739435832, −5.42865654179937754919858830837, −4.89248644221476904982666450124, −4.01664712670402337909545823196, −2.98715823273683233786617758464, −2.22893314956303996702896408026, −1.25723939815807264195696589803, 0, 1.25723939815807264195696589803, 2.22893314956303996702896408026, 2.98715823273683233786617758464, 4.01664712670402337909545823196, 4.89248644221476904982666450124, 5.42865654179937754919858830837, 6.17820793698472085055739435832, 6.59887555103524585392383492632, 7.57217738192382147009034730617

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