Properties

Label 2-7728-1.1-c1-0-77
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 − 3.94·5-s − 7-s + 9-s − 1.72·11-s + 2.21·13-s − 3.94·15-s − 5.14·17-s + 1.72·19-s − 21-s + 23-s + 10.5·25-s + 27-s + 3.01·29-s + 8.20·31-s − 1.72·33-s + 3.94·35-s + 7.61·37-s + 2.21·39-s + 1.14·41-s − 7.22·43-s − 3.94·45-s − 7.34·47-s + 49-s − 5.14·51-s − 4.21·53-s + 6.82·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.76·5-s − 0.377·7-s + 0.333·9-s − 0.521·11-s + 0.614·13-s − 1.01·15-s − 1.24·17-s + 0.396·19-s − 0.218·21-s + 0.208·23-s + 2.11·25-s + 0.192·27-s + 0.558·29-s + 1.47·31-s − 0.301·33-s + 0.666·35-s + 1.25·37-s + 0.354·39-s + 0.179·41-s − 1.10·43-s − 0.588·45-s − 1.07·47-s + 0.142·49-s − 0.720·51-s − 0.578·53-s + 0.919·55-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: Trivial
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 \( 1 + 3.94T + 5T^{2} \)
11 \( 1 + 1.72T + 11T^{2} \)
13 \( 1 - 2.21T + 13T^{2} \)
17 \( 1 + 5.14T + 17T^{2} \)
19 \( 1 - 1.72T + 19T^{2} \)
29 \( 1 - 3.01T + 29T^{2} \)
31 \( 1 - 8.20T + 31T^{2} \)
37 \( 1 - 7.61T + 37T^{2} \)
41 \( 1 - 1.14T + 41T^{2} \)
43 \( 1 + 7.22T + 43T^{2} \)
47 \( 1 + 7.34T + 47T^{2} \)
53 \( 1 + 4.21T + 53T^{2} \)
59 \( 1 - 1.33T + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 - 0.0825T + 67T^{2} \)
71 \( 1 - 5.53T + 71T^{2} \)
73 \( 1 + 8.33T + 73T^{2} \)
79 \( 1 + 8.20T + 79T^{2} \)
83 \( 1 - 3.21T + 83T^{2} \)
89 \( 1 - 9.45T + 89T^{2} \)
97 \( 1 + 8.33T + 97T^{2} \)
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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

−7.75319712571790788886950948943, −6.79150490468149509658498424893, −6.51396687852448522575213573882, −5.20131257236212580212902161350, −4.43846301463721440147027728352, −3.94739389278496644838021107248, −3.12737881616901594223540217925, −2.56488344005516336837722214407, −1.09889441847995240216728185159, 0, 1.09889441847995240216728185159, 2.56488344005516336837722214407, 3.12737881616901594223540217925, 3.94739389278496644838021107248, 4.43846301463721440147027728352, 5.20131257236212580212902161350, 6.51396687852448522575213573882, 6.79150490468149509658498424893, 7.75319712571790788886950948943

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