Properties

Label 2-7728-1.1-c1-0-107
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 + 2.57·5-s + 7-s + 9-s − 4.81·11-s + 4.91·13-s − 2.57·15-s − 7.21·17-s − 2.33·19-s − 21-s + 23-s + 1.64·25-s − 27-s + 3.10·29-s + 1.21·31-s + 4.81·33-s + 2.57·35-s + 0.223·37-s − 4.91·39-s + 8.25·41-s − 7.31·43-s + 2.57·45-s − 8.11·47-s + 49-s + 7.21·51-s − 1.13·53-s − 12.4·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.15·5-s + 0.377·7-s + 0.333·9-s − 1.45·11-s + 1.36·13-s − 0.665·15-s − 1.75·17-s − 0.535·19-s − 0.218·21-s + 0.208·23-s + 0.328·25-s − 0.192·27-s + 0.576·29-s + 0.218·31-s + 0.838·33-s + 0.435·35-s + 0.0368·37-s − 0.786·39-s + 1.28·41-s − 1.11·43-s + 0.384·45-s − 1.18·47-s + 0.142·49-s + 1.01·51-s − 0.156·53-s − 1.67·55-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 \( 1 - 2.57T + 5T^{2} \)
11 \( 1 + 4.81T + 11T^{2} \)
13 \( 1 - 4.91T + 13T^{2} \)
17 \( 1 + 7.21T + 17T^{2} \)
19 \( 1 + 2.33T + 19T^{2} \)
29 \( 1 - 3.10T + 29T^{2} \)
31 \( 1 - 1.21T + 31T^{2} \)
37 \( 1 - 0.223T + 37T^{2} \)
41 \( 1 - 8.25T + 41T^{2} \)
43 \( 1 + 7.31T + 43T^{2} \)
47 \( 1 + 8.11T + 47T^{2} \)
53 \( 1 + 1.13T + 53T^{2} \)
59 \( 1 - 9.56T + 59T^{2} \)
61 \( 1 + 14.1T + 61T^{2} \)
67 \( 1 + 6.01T + 67T^{2} \)
71 \( 1 - 4.75T + 71T^{2} \)
73 \( 1 + 9.70T + 73T^{2} \)
79 \( 1 + 5.87T + 79T^{2} \)
83 \( 1 + 4.60T + 83T^{2} \)
89 \( 1 - 7.95T + 89T^{2} \)
97 \( 1 + 4.43T + 97T^{2} \)
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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

−7.47817382847380088337441038463, −6.49561144184635005707210527498, −6.21932091351434082424055980273, −5.45835838900808342375065057205, −4.82365644963971572518801292763, −4.14626059388879338457055550121, −2.90790531659365636493665943365, −2.15793566670971130851909772768, −1.36787069694414502873132455750, 0, 1.36787069694414502873132455750, 2.15793566670971130851909772768, 2.90790531659365636493665943365, 4.14626059388879338457055550121, 4.82365644963971572518801292763, 5.45835838900808342375065057205, 6.21932091351434082424055980273, 6.49561144184635005707210527498, 7.47817382847380088337441038463

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