Properties

Label 2-28e2-1.1-c1-0-2
Degree $2$
Conductor $784$
Sign $1$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 3·9-s + 4·11-s − 2·13-s + 6·17-s + 8·19-s − 25-s + 6·29-s + 8·31-s − 2·37-s − 2·41-s + 4·43-s + 6·45-s − 8·47-s + 6·53-s − 8·55-s + 6·61-s + 4·65-s + 4·67-s + 8·71-s − 10·73-s − 16·79-s + 9·81-s + 8·83-s − 12·85-s + 6·89-s − 16·95-s + ⋯
L(s)  = 1  − 0.894·5-s − 9-s + 1.20·11-s − 0.554·13-s + 1.45·17-s + 1.83·19-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.328·37-s − 0.312·41-s + 0.609·43-s + 0.894·45-s − 1.16·47-s + 0.824·53-s − 1.07·55-s + 0.768·61-s + 0.496·65-s + 0.488·67-s + 0.949·71-s − 1.17·73-s − 1.80·79-s + 81-s + 0.878·83-s − 1.30·85-s + 0.635·89-s − 1.64·95-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{784} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.322192770\)
\(L(\frac12)\) \(\approx\) \(1.322192770\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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

−10.15969982113999416146742248886, −9.502244133602489113553033388660, −8.459138850523211127936437645267, −7.80607013998257452679138271336, −6.92383462522053566355627597037, −5.84465028770243608573987581113, −4.90615311225424089857668766077, −3.69860481691342078568866024595, −2.92766858317331148516836536443, −0.988078077776060913977316087411, 0.988078077776060913977316087411, 2.92766858317331148516836536443, 3.69860481691342078568866024595, 4.90615311225424089857668766077, 5.84465028770243608573987581113, 6.92383462522053566355627597037, 7.80607013998257452679138271336, 8.459138850523211127936437645267, 9.502244133602489113553033388660, 10.15969982113999416146742248886

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