Properties

Label 2-7728-1.1-c1-0-41
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2.48·5-s + 7-s + 9-s + 5.58·11-s − 5.15·13-s − 2.48·15-s + 3.97·17-s + 5.58·19-s + 21-s + 23-s + 1.18·25-s + 27-s − 1.84·29-s + 8.60·31-s + 5.58·33-s − 2.48·35-s + 2.80·37-s − 5.15·39-s − 4.16·41-s − 4.11·43-s − 2.48·45-s + 4.18·47-s + 49-s + 3.97·51-s − 8.87·53-s − 13.8·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.11·5-s + 0.377·7-s + 0.333·9-s + 1.68·11-s − 1.42·13-s − 0.641·15-s + 0.964·17-s + 1.28·19-s + 0.218·21-s + 0.208·23-s + 0.236·25-s + 0.192·27-s − 0.342·29-s + 1.54·31-s + 0.972·33-s − 0.420·35-s + 0.460·37-s − 0.825·39-s − 0.650·41-s − 0.627·43-s − 0.370·45-s + 0.611·47-s + 0.142·49-s + 0.556·51-s − 1.21·53-s − 1.87·55-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

\(L(1)\) \(\approx\) \(2.421428139\)
\(L(\frac12)\) \(\approx\) \(2.421428139\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.48T + 5T^{2} \)
11 \( 1 - 5.58T + 11T^{2} \)
13 \( 1 + 5.15T + 13T^{2} \)
17 \( 1 - 3.97T + 17T^{2} \)
19 \( 1 - 5.58T + 19T^{2} \)
29 \( 1 + 1.84T + 29T^{2} \)
31 \( 1 - 8.60T + 31T^{2} \)
37 \( 1 - 2.80T + 37T^{2} \)
41 \( 1 + 4.16T + 41T^{2} \)
43 \( 1 + 4.11T + 43T^{2} \)
47 \( 1 - 4.18T + 47T^{2} \)
53 \( 1 + 8.87T + 53T^{2} \)
59 \( 1 - 2.74T + 59T^{2} \)
61 \( 1 + 6.89T + 61T^{2} \)
67 \( 1 - 7.59T + 67T^{2} \)
71 \( 1 + 5.88T + 71T^{2} \)
73 \( 1 - 9.31T + 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 - 5.08T + 89T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.87930141336268944977639252907, −7.27136949987159736189445740980, −6.78196775894662573292308561655, −5.74228554132892938683443100603, −4.79925961493551054944963260729, −4.29221482061320068254410893012, −3.47188849760518258688070294402, −2.92726095925159181928255394494, −1.70263354059082544307214306915, −0.792833659183510084243003157604, 0.792833659183510084243003157604, 1.70263354059082544307214306915, 2.92726095925159181928255394494, 3.47188849760518258688070294402, 4.29221482061320068254410893012, 4.79925961493551054944963260729, 5.74228554132892938683443100603, 6.78196775894662573292308561655, 7.27136949987159736189445740980, 7.87930141336268944977639252907

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