Properties

Label 2-7728-1.1-c1-0-125
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 + 1.37·5-s + 7-s + 9-s − 4·11-s + 1.37·13-s + 1.37·15-s − 4.74·17-s − 4·19-s + 21-s + 23-s − 3.11·25-s + 27-s + 9.37·29-s − 6.74·31-s − 4·33-s + 1.37·35-s + 2.62·37-s + 1.37·39-s − 8.11·41-s − 6.11·43-s + 1.37·45-s − 4.62·47-s + 49-s − 4.74·51-s + 4.74·53-s − 5.48·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.613·5-s + 0.377·7-s + 0.333·9-s − 1.20·11-s + 0.380·13-s + 0.354·15-s − 1.15·17-s − 0.917·19-s + 0.218·21-s + 0.208·23-s − 0.623·25-s + 0.192·27-s + 1.74·29-s − 1.21·31-s − 0.696·33-s + 0.231·35-s + 0.431·37-s + 0.219·39-s − 1.26·41-s − 0.932·43-s + 0.204·45-s − 0.675·47-s + 0.142·49-s − 0.664·51-s + 0.651·53-s − 0.740·55-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 \( 1 - 1.37T + 5T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 1.37T + 13T^{2} \)
17 \( 1 + 4.74T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
29 \( 1 - 9.37T + 29T^{2} \)
31 \( 1 + 6.74T + 31T^{2} \)
37 \( 1 - 2.62T + 37T^{2} \)
41 \( 1 + 8.11T + 41T^{2} \)
43 \( 1 + 6.11T + 43T^{2} \)
47 \( 1 + 4.62T + 47T^{2} \)
53 \( 1 - 4.74T + 53T^{2} \)
59 \( 1 - 2.74T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 - 7.48T + 89T^{2} \)
97 \( 1 + 9.37T + 97T^{2} \)
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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

−7.61281398134608590732092216128, −6.79838029264534286585467225560, −6.22024323730082386104415434298, −5.32637321486472345150759045410, −4.72339418726495577628536825309, −3.94119327514708601803431821617, −2.92626084659497970937035985799, −2.26307948419910956403881145132, −1.54920649876814086868171167078, 0, 1.54920649876814086868171167078, 2.26307948419910956403881145132, 2.92626084659497970937035985799, 3.94119327514708601803431821617, 4.72339418726495577628536825309, 5.32637321486472345150759045410, 6.22024323730082386104415434298, 6.79838029264534286585467225560, 7.61281398134608590732092216128

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