Properties

Label 2-7728-1.1-c1-0-65
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 − 4.30·5-s + 7-s + 9-s + 5·11-s − 1.30·13-s + 4.30·15-s + 1.60·17-s − 5.60·19-s − 21-s − 23-s + 13.5·25-s − 27-s − 8.21·29-s − 3·31-s − 5·33-s − 4.30·35-s − 9·37-s + 1.30·39-s + 2.21·41-s + 12.5·43-s − 4.30·45-s + 1.39·47-s + 49-s − 1.60·51-s + 5.51·53-s − 21.5·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.92·5-s + 0.377·7-s + 0.333·9-s + 1.50·11-s − 0.361·13-s + 1.11·15-s + 0.389·17-s − 1.28·19-s − 0.218·21-s − 0.208·23-s + 2.70·25-s − 0.192·27-s − 1.52·29-s − 0.538·31-s − 0.870·33-s − 0.727·35-s − 1.47·37-s + 0.208·39-s + 0.345·41-s + 1.90·43-s − 0.641·45-s + 0.203·47-s + 0.142·49-s − 0.224·51-s + 0.757·53-s − 2.90·55-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 \( 1 + 4.30T + 5T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 + 1.30T + 13T^{2} \)
17 \( 1 - 1.60T + 17T^{2} \)
19 \( 1 + 5.60T + 19T^{2} \)
29 \( 1 + 8.21T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 + 9T + 37T^{2} \)
41 \( 1 - 2.21T + 41T^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 - 1.39T + 47T^{2} \)
53 \( 1 - 5.51T + 53T^{2} \)
59 \( 1 - 6.90T + 59T^{2} \)
61 \( 1 + 11.9T + 61T^{2} \)
67 \( 1 - 1.09T + 67T^{2} \)
71 \( 1 - 9.90T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 - T + 79T^{2} \)
83 \( 1 - 1.60T + 83T^{2} \)
89 \( 1 - 0.0916T + 89T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−7.42735501432499161314849484389, −6.99938847239455516893047106437, −6.23569500465333154267857762879, −5.32874608905710848336060931194, −4.49266501787099364812786350392, −3.91889671583570099263211873335, −3.56730753411106306693623741931, −2.14052998086005200911494611372, −1.00012505549975965217365163366, 0, 1.00012505549975965217365163366, 2.14052998086005200911494611372, 3.56730753411106306693623741931, 3.91889671583570099263211873335, 4.49266501787099364812786350392, 5.32874608905710848336060931194, 6.23569500465333154267857762879, 6.99938847239455516893047106437, 7.42735501432499161314849484389

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