Properties

Label 2-7728-1.1-c1-0-66
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4·5-s + 7-s + 9-s + 5·11-s − 2·13-s − 4·15-s + 5·19-s − 21-s + 23-s + 11·25-s − 27-s − 2·29-s − 6·31-s − 5·33-s + 4·35-s + 6·37-s + 2·39-s + 5·41-s − 8·43-s + 4·45-s + 9·47-s + 49-s + 9·53-s + 20·55-s − 5·57-s − 9·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s + 1.50·11-s − 0.554·13-s − 1.03·15-s + 1.14·19-s − 0.218·21-s + 0.208·23-s + 11/5·25-s − 0.192·27-s − 0.371·29-s − 1.07·31-s − 0.870·33-s + 0.676·35-s + 0.986·37-s + 0.320·39-s + 0.780·41-s − 1.21·43-s + 0.596·45-s + 1.31·47-s + 1/7·49-s + 1.23·53-s + 2.69·55-s − 0.662·57-s − 1.17·59-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: \(0\)
Selberg data: \((2,\ 7728,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.097549591\)
\(L(\frac12)\) \(\approx\) \(3.097549591\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−7.63158150513750950337647130419, −7.05817953077553731244250788627, −6.30271385755438495186139887277, −5.82636434969157029969719859702, −5.21400971049369667954496425971, −4.52798276521074803880719468746, −3.53520334126760659677282274992, −2.46725105075483857162906337780, −1.65098046847587676418022247486, −1.00050204410442716469698004603, 1.00050204410442716469698004603, 1.65098046847587676418022247486, 2.46725105075483857162906337780, 3.53520334126760659677282274992, 4.52798276521074803880719468746, 5.21400971049369667954496425971, 5.82636434969157029969719859702, 6.30271385755438495186139887277, 7.05817953077553731244250788627, 7.63158150513750950337647130419

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