Properties

Label 2-7728-1.1-c1-0-16
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 − 1.52·5-s + 7-s + 9-s − 1.86·11-s + 0.604·13-s + 1.52·15-s − 2.92·17-s + 1.86·19-s − 21-s + 23-s − 2.66·25-s − 27-s + 7.19·29-s + 5.86·31-s + 1.86·33-s − 1.52·35-s − 3.19·37-s − 0.604·39-s − 6.65·41-s + 5.66·43-s − 1.52·45-s − 10.7·47-s + 49-s + 2.92·51-s − 1.52·53-s + 2.85·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.683·5-s + 0.377·7-s + 0.333·9-s − 0.562·11-s + 0.167·13-s + 0.394·15-s − 0.709·17-s + 0.428·19-s − 0.218·21-s + 0.208·23-s − 0.532·25-s − 0.192·27-s + 1.33·29-s + 1.05·31-s + 0.324·33-s − 0.258·35-s − 0.524·37-s − 0.0968·39-s − 1.03·41-s + 0.863·43-s − 0.227·45-s − 1.57·47-s + 0.142·49-s + 0.409·51-s − 0.210·53-s + 0.384·55-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

\(L(1)\) \(\approx\) \(1.108144082\)
\(L(\frac12)\) \(\approx\) \(1.108144082\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.52T + 5T^{2} \)
11 \( 1 + 1.86T + 11T^{2} \)
13 \( 1 - 0.604T + 13T^{2} \)
17 \( 1 + 2.92T + 17T^{2} \)
19 \( 1 - 1.86T + 19T^{2} \)
29 \( 1 - 7.19T + 29T^{2} \)
31 \( 1 - 5.86T + 31T^{2} \)
37 \( 1 + 3.19T + 37T^{2} \)
41 \( 1 + 6.65T + 41T^{2} \)
43 \( 1 - 5.66T + 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + 1.52T + 53T^{2} \)
59 \( 1 - 6.31T + 59T^{2} \)
61 \( 1 + 3.66T + 61T^{2} \)
67 \( 1 + 5.37T + 67T^{2} \)
71 \( 1 + 15.2T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 + 9.19T + 79T^{2} \)
83 \( 1 - 0.924T + 83T^{2} \)
89 \( 1 - 7.39T + 89T^{2} \)
97 \( 1 - 5.98T + 97T^{2} \)
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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

−7.906027758133638706035286735519, −7.14987501411402489783279258206, −6.50956496658022966666959761432, −5.78720895441251495112741824511, −4.87293144868399731247894068704, −4.54820595190026370507019851860, −3.58005816341294718145395960014, −2.74258983520197288409493465291, −1.66021402336329761698448652519, −0.54634687373185361043925360645, 0.54634687373185361043925360645, 1.66021402336329761698448652519, 2.74258983520197288409493465291, 3.58005816341294718145395960014, 4.54820595190026370507019851860, 4.87293144868399731247894068704, 5.78720895441251495112741824511, 6.50956496658022966666959761432, 7.14987501411402489783279258206, 7.906027758133638706035286735519

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