Properties

Label 2-7728-1.1-c1-0-35
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 − 2.73·5-s + 7-s + 9-s + 2.50·11-s + 1.48·13-s − 2.73·15-s + 0.902·17-s − 2.50·19-s + 21-s − 23-s + 2.48·25-s + 27-s + 6.68·29-s − 1.09·31-s + 2.50·33-s − 2.73·35-s + 9.91·37-s + 1.48·39-s − 2.30·41-s − 5.83·43-s − 2.73·45-s + 6.74·47-s + 49-s + 0.902·51-s − 4.91·53-s − 6.85·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.22·5-s + 0.377·7-s + 0.333·9-s + 0.755·11-s + 0.411·13-s − 0.706·15-s + 0.218·17-s − 0.574·19-s + 0.218·21-s − 0.208·23-s + 0.497·25-s + 0.192·27-s + 1.24·29-s − 0.197·31-s + 0.435·33-s − 0.462·35-s + 1.62·37-s + 0.237·39-s − 0.360·41-s − 0.889·43-s − 0.407·45-s + 0.984·47-s + 0.142·49-s + 0.126·51-s − 0.675·53-s − 0.923·55-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

\(L(1)\) \(\approx\) \(2.181539752\)
\(L(\frac12)\) \(\approx\) \(2.181539752\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.73T + 5T^{2} \)
11 \( 1 - 2.50T + 11T^{2} \)
13 \( 1 - 1.48T + 13T^{2} \)
17 \( 1 - 0.902T + 17T^{2} \)
19 \( 1 + 2.50T + 19T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 + 1.09T + 31T^{2} \)
37 \( 1 - 9.91T + 37T^{2} \)
41 \( 1 + 2.30T + 41T^{2} \)
43 \( 1 + 5.83T + 43T^{2} \)
47 \( 1 - 6.74T + 47T^{2} \)
53 \( 1 + 4.91T + 53T^{2} \)
59 \( 1 - 7.32T + 59T^{2} \)
61 \( 1 + 1.00T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 - 0.362T + 71T^{2} \)
73 \( 1 + 5.34T + 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 + 7.59T + 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 - 13.9T + 97T^{2} \)
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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

−7.83413139288207986680617646234, −7.40478728232605161662095130662, −6.56540528591838751848683844420, −5.90877058063608895103734391268, −4.70684107124115494409635396712, −4.26302480931936082470429669681, −3.59217100121006549285649791427, −2.84422429082652816608461463835, −1.75571453526589894719714173959, −0.73633593168246341386534192271, 0.73633593168246341386534192271, 1.75571453526589894719714173959, 2.84422429082652816608461463835, 3.59217100121006549285649791427, 4.26302480931936082470429669681, 4.70684107124115494409635396712, 5.90877058063608895103734391268, 6.56540528591838751848683844420, 7.40478728232605161662095130662, 7.83413139288207986680617646234

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