Properties

Label 2-7728-1.1-c1-0-23
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 + 1.11·5-s − 7-s + 9-s + 3.94·11-s − 5.47·13-s − 1.11·15-s − 0.926·17-s + 2.28·19-s + 21-s + 23-s − 3.75·25-s − 27-s + 2.58·29-s − 8.81·31-s − 3.94·33-s − 1.11·35-s + 1.87·37-s + 5.47·39-s − 4.28·41-s + 9.70·43-s + 1.11·45-s + 7.81·47-s + 49-s + 0.926·51-s + 4.70·53-s + 4.39·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.498·5-s − 0.377·7-s + 0.333·9-s + 1.18·11-s − 1.51·13-s − 0.287·15-s − 0.224·17-s + 0.524·19-s + 0.218·21-s + 0.208·23-s − 0.751·25-s − 0.192·27-s + 0.480·29-s − 1.58·31-s − 0.686·33-s − 0.188·35-s + 0.307·37-s + 0.876·39-s − 0.669·41-s + 1.47·43-s + 0.166·45-s + 1.14·47-s + 0.142·49-s + 0.129·51-s + 0.645·53-s + 0.593·55-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

\(L(1)\) \(\approx\) \(1.516453905\)
\(L(\frac12)\) \(\approx\) \(1.516453905\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.11T + 5T^{2} \)
11 \( 1 - 3.94T + 11T^{2} \)
13 \( 1 + 5.47T + 13T^{2} \)
17 \( 1 + 0.926T + 17T^{2} \)
19 \( 1 - 2.28T + 19T^{2} \)
29 \( 1 - 2.58T + 29T^{2} \)
31 \( 1 + 8.81T + 31T^{2} \)
37 \( 1 - 1.87T + 37T^{2} \)
41 \( 1 + 4.28T + 41T^{2} \)
43 \( 1 - 9.70T + 43T^{2} \)
47 \( 1 - 7.81T + 47T^{2} \)
53 \( 1 - 4.70T + 53T^{2} \)
59 \( 1 - 6.93T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 - 8.77T + 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 + 7.87T + 73T^{2} \)
79 \( 1 + 8.81T + 79T^{2} \)
83 \( 1 + 7.04T + 83T^{2} \)
89 \( 1 - 0.0411T + 89T^{2} \)
97 \( 1 - 16.5T + 97T^{2} \)
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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

−7.48981699592073499232967562495, −7.23893526855482389707831202479, −6.41907459066623246056143349431, −5.79733422192656310159452453613, −5.16975667302840806631014138896, −4.34907105107315172499868487063, −3.63940091199571034108160275026, −2.56937446840366153119703219927, −1.76870898015729211768150745502, −0.63087884767814348039292938058, 0.63087884767814348039292938058, 1.76870898015729211768150745502, 2.56937446840366153119703219927, 3.63940091199571034108160275026, 4.34907105107315172499868487063, 5.16975667302840806631014138896, 5.79733422192656310159452453613, 6.41907459066623246056143349431, 7.23893526855482389707831202479, 7.48981699592073499232967562495

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