Properties

Label 2-7728-1.1-c1-0-14
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2.37·5-s − 7-s + 9-s − 3.37·11-s − 0.372·13-s − 2.37·15-s + 4·17-s − 7.37·19-s − 21-s − 23-s + 0.627·25-s + 27-s − 0.372·29-s + 6·31-s − 3.37·33-s + 2.37·35-s − 8.37·37-s − 0.372·39-s + 5.74·41-s + 2.37·43-s − 2.37·45-s + 7.74·47-s + 49-s + 4·51-s − 10.1·53-s + 8·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.06·5-s − 0.377·7-s + 0.333·9-s − 1.01·11-s − 0.103·13-s − 0.612·15-s + 0.970·17-s − 1.69·19-s − 0.218·21-s − 0.208·23-s + 0.125·25-s + 0.192·27-s − 0.0691·29-s + 1.07·31-s − 0.587·33-s + 0.400·35-s − 1.37·37-s − 0.0596·39-s + 0.897·41-s + 0.361·43-s − 0.353·45-s + 1.12·47-s + 0.142·49-s + 0.560·51-s − 1.38·53-s + 1.07·55-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: \(0\)
Selberg data: \((2,\ 7728,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.209473298\)
\(L(\frac12)\) \(\approx\) \(1.209473298\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.37T + 5T^{2} \)
11 \( 1 + 3.37T + 11T^{2} \)
13 \( 1 + 0.372T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 7.37T + 19T^{2} \)
29 \( 1 + 0.372T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 8.37T + 37T^{2} \)
41 \( 1 - 5.74T + 41T^{2} \)
43 \( 1 - 2.37T + 43T^{2} \)
47 \( 1 - 7.74T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + 9.37T + 59T^{2} \)
61 \( 1 + 2.62T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 - 12T + 73T^{2} \)
79 \( 1 - 2.74T + 79T^{2} \)
83 \( 1 + 2.74T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 - 5.11T + 97T^{2} \)
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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

−7.80406127815212048084138515442, −7.48600622705977314119406057216, −6.54007521929491323703797781594, −5.85344377678486701385229915042, −4.85399504610240050414086337419, −4.23007640608338524492251484060, −3.48541038078183103078959796312, −2.81754412317450386034233329442, −1.93928301331908762596962685366, −0.50953969450420984073089415385, 0.50953969450420984073089415385, 1.93928301331908762596962685366, 2.81754412317450386034233329442, 3.48541038078183103078959796312, 4.23007640608338524492251484060, 4.85399504610240050414086337419, 5.85344377678486701385229915042, 6.54007521929491323703797781594, 7.48600622705977314119406057216, 7.80406127815212048084138515442

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