Properties

Label 2-7728-1.1-c1-0-103
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 − 0.681·5-s + 7-s + 9-s + 3.71·11-s + 3.66·13-s + 0.681·15-s + 2.21·17-s − 4.34·19-s − 21-s + 23-s − 4.53·25-s − 27-s − 4.63·29-s − 8.21·31-s − 3.71·33-s − 0.681·35-s + 3.93·37-s − 3.66·39-s − 5.99·41-s − 5.16·43-s − 0.681·45-s − 6.41·47-s + 49-s − 2.21·51-s − 3.60·53-s − 2.53·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.304·5-s + 0.377·7-s + 0.333·9-s + 1.11·11-s + 1.01·13-s + 0.176·15-s + 0.537·17-s − 0.997·19-s − 0.218·21-s + 0.208·23-s − 0.907·25-s − 0.192·27-s − 0.860·29-s − 1.47·31-s − 0.646·33-s − 0.115·35-s + 0.646·37-s − 0.587·39-s − 0.936·41-s − 0.787·43-s − 0.101·45-s − 0.935·47-s + 0.142·49-s − 0.310·51-s − 0.494·53-s − 0.341·55-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: \(1\)
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 \( 1 + 0.681T + 5T^{2} \)
11 \( 1 - 3.71T + 11T^{2} \)
13 \( 1 - 3.66T + 13T^{2} \)
17 \( 1 - 2.21T + 17T^{2} \)
19 \( 1 + 4.34T + 19T^{2} \)
29 \( 1 + 4.63T + 29T^{2} \)
31 \( 1 + 8.21T + 31T^{2} \)
37 \( 1 - 3.93T + 37T^{2} \)
41 \( 1 + 5.99T + 41T^{2} \)
43 \( 1 + 5.16T + 43T^{2} \)
47 \( 1 + 6.41T + 47T^{2} \)
53 \( 1 + 3.60T + 53T^{2} \)
59 \( 1 + 6.83T + 59T^{2} \)
61 \( 1 + 4.71T + 61T^{2} \)
67 \( 1 - 2.96T + 67T^{2} \)
71 \( 1 + 3.12T + 71T^{2} \)
73 \( 1 - 10.2T + 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 - 1.88T + 89T^{2} \)
97 \( 1 - 7.33T + 97T^{2} \)
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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

−7.54316890499266664487058019619, −6.66731734643050189299187499755, −6.19705035906083131208765395670, −5.48790034776395795929884703886, −4.67329642749081535122339221109, −3.85420856909961788580721791298, −3.45597522361879496009887486711, −1.93994389665512026441717492821, −1.31847790525187226101400965227, 0, 1.31847790525187226101400965227, 1.93994389665512026441717492821, 3.45597522361879496009887486711, 3.85420856909961788580721791298, 4.67329642749081535122339221109, 5.48790034776395795929884703886, 6.19705035906083131208765395670, 6.66731734643050189299187499755, 7.54316890499266664487058019619

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