Properties

Label 2-7728-1.1-c1-0-95
Degree $2$
Conductor $7728$
Sign $-1$
Analytic cond. $61.7083$
Root an. cond. $7.85546$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·5-s + 7-s + 9-s − 13-s − 3·15-s − 4·17-s + 21-s − 23-s + 4·25-s + 27-s + 5·29-s + 10·31-s − 3·35-s − 3·37-s − 39-s − 11·41-s + 9·43-s − 3·45-s + 3·47-s + 49-s − 4·51-s + 12·53-s − 10·59-s − 10·61-s + 63-s + 3·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.277·13-s − 0.774·15-s − 0.970·17-s + 0.218·21-s − 0.208·23-s + 4/5·25-s + 0.192·27-s + 0.928·29-s + 1.79·31-s − 0.507·35-s − 0.493·37-s − 0.160·39-s − 1.71·41-s + 1.37·43-s − 0.447·45-s + 0.437·47-s + 1/7·49-s − 0.560·51-s + 1.64·53-s − 1.30·59-s − 1.28·61-s + 0.125·63-s + 0.372·65-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: yes
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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−7.59965267583394120895466385026, −7.00217750434419845410448942993, −6.33582082496062884460268580844, −5.21569438121451725456142195156, −4.38789318063603582788983743599, −4.10721408376758978447161596513, −3.08400932814634790744861853224, −2.43378750100698839758206508134, −1.23347331354479212409499920845, 0, 1.23347331354479212409499920845, 2.43378750100698839758206508134, 3.08400932814634790744861853224, 4.10721408376758978447161596513, 4.38789318063603582788983743599, 5.21569438121451725456142195156, 6.33582082496062884460268580844, 7.00217750434419845410448942993, 7.59965267583394120895466385026

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