Properties

Label 2-28e2-1.1-c5-0-37
Degree $2$
Conductor $784$
Sign $1$
Analytic cond. $125.740$
Root an. cond. $11.2134$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 6.89·3-s + 48.4·5-s − 195.·9-s + 163.·11-s + 120.·13-s + 334.·15-s − 78.1·17-s − 2.26e3·19-s + 2.45e3·23-s − 776.·25-s − 3.02e3·27-s + 6.98e3·29-s + 2.79e3·31-s + 1.12e3·33-s + 9.45e3·37-s + 832.·39-s − 1.00e4·41-s + 6.93e3·43-s − 9.47e3·45-s + 1.16e3·47-s − 538.·51-s + 8.56e3·53-s + 7.92e3·55-s − 1.56e4·57-s + 6.22e3·59-s + 4.19e4·61-s + 5.85e3·65-s + ⋯
L(s)  = 1  + 0.442·3-s + 0.866·5-s − 0.804·9-s + 0.407·11-s + 0.198·13-s + 0.383·15-s − 0.0656·17-s − 1.43·19-s + 0.966·23-s − 0.248·25-s − 0.797·27-s + 1.54·29-s + 0.522·31-s + 0.180·33-s + 1.13·37-s + 0.0876·39-s − 0.937·41-s + 0.571·43-s − 0.697·45-s + 0.0769·47-s − 0.0290·51-s + 0.418·53-s + 0.353·55-s − 0.636·57-s + 0.232·59-s + 1.44·61-s + 0.171·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(125.740\)
Root analytic conductor: \(11.2134\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.990362635\)
\(L(\frac12)\) \(\approx\) \(2.990362635\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 - 6.89T + 243T^{2} \)
5 \( 1 - 48.4T + 3.12e3T^{2} \)
11 \( 1 - 163.T + 1.61e5T^{2} \)
13 \( 1 - 120.T + 3.71e5T^{2} \)
17 \( 1 + 78.1T + 1.41e6T^{2} \)
19 \( 1 + 2.26e3T + 2.47e6T^{2} \)
23 \( 1 - 2.45e3T + 6.43e6T^{2} \)
29 \( 1 - 6.98e3T + 2.05e7T^{2} \)
31 \( 1 - 2.79e3T + 2.86e7T^{2} \)
37 \( 1 - 9.45e3T + 6.93e7T^{2} \)
41 \( 1 + 1.00e4T + 1.15e8T^{2} \)
43 \( 1 - 6.93e3T + 1.47e8T^{2} \)
47 \( 1 - 1.16e3T + 2.29e8T^{2} \)
53 \( 1 - 8.56e3T + 4.18e8T^{2} \)
59 \( 1 - 6.22e3T + 7.14e8T^{2} \)
61 \( 1 - 4.19e4T + 8.44e8T^{2} \)
67 \( 1 + 1.81e3T + 1.35e9T^{2} \)
71 \( 1 - 5.68e4T + 1.80e9T^{2} \)
73 \( 1 - 4.42e4T + 2.07e9T^{2} \)
79 \( 1 + 3.49e4T + 3.07e9T^{2} \)
83 \( 1 + 3.96e4T + 3.93e9T^{2} \)
89 \( 1 - 1.26e5T + 5.58e9T^{2} \)
97 \( 1 - 1.45e5T + 8.58e9T^{2} \)
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

−9.451829956363657024653924548405, −8.693338469435825687156315515797, −8.099598967670141187943732271770, −6.72950774177481179857760282477, −6.14474069399899338652475443250, −5.14689426228218125915424948791, −4.03419559650972499618254299015, −2.83221284806293859105693043651, −2.07029295629872772380746580973, −0.77119680208208657256980920474, 0.77119680208208657256980920474, 2.07029295629872772380746580973, 2.83221284806293859105693043651, 4.03419559650972499618254299015, 5.14689426228218125915424948791, 6.14474069399899338652475443250, 6.72950774177481179857760282477, 8.099598967670141187943732271770, 8.693338469435825687156315515797, 9.451829956363657024653924548405

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