Properties

Degree $2$
Conductor $784$
Sign $-1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 2·5-s − 11·9-s + 44·11-s − 22·13-s − 8·15-s − 50·17-s + 44·19-s + 56·23-s − 121·25-s + 152·27-s + 198·29-s − 160·31-s − 176·33-s − 162·37-s + 88·39-s + 198·41-s − 52·43-s − 22·45-s + 528·47-s + 200·51-s − 242·53-s + 88·55-s − 176·57-s − 668·59-s − 550·61-s − 44·65-s + ⋯
L(s)  = 1  − 0.769·3-s + 0.178·5-s − 0.407·9-s + 1.20·11-s − 0.469·13-s − 0.137·15-s − 0.713·17-s + 0.531·19-s + 0.507·23-s − 0.967·25-s + 1.08·27-s + 1.26·29-s − 0.926·31-s − 0.928·33-s − 0.719·37-s + 0.361·39-s + 0.754·41-s − 0.184·43-s − 0.0728·45-s + 1.63·47-s + 0.549·51-s − 0.627·53-s + 0.215·55-s − 0.408·57-s − 1.47·59-s − 1.15·61-s − 0.0839·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(3\)
Character: $\chi_{784} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 784,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 4 T + p^{3} T^{2} \)
5 \( 1 - 2 T + p^{3} T^{2} \)
11 \( 1 - 4 p T + p^{3} T^{2} \)
13 \( 1 + 22 T + p^{3} T^{2} \)
17 \( 1 + 50 T + p^{3} T^{2} \)
19 \( 1 - 44 T + p^{3} T^{2} \)
23 \( 1 - 56 T + p^{3} T^{2} \)
29 \( 1 - 198 T + p^{3} T^{2} \)
31 \( 1 + 160 T + p^{3} T^{2} \)
37 \( 1 + 162 T + p^{3} T^{2} \)
41 \( 1 - 198 T + p^{3} T^{2} \)
43 \( 1 + 52 T + p^{3} T^{2} \)
47 \( 1 - 528 T + p^{3} T^{2} \)
53 \( 1 + 242 T + p^{3} T^{2} \)
59 \( 1 + 668 T + p^{3} T^{2} \)
61 \( 1 + 550 T + p^{3} T^{2} \)
67 \( 1 + 188 T + p^{3} T^{2} \)
71 \( 1 + 728 T + p^{3} T^{2} \)
73 \( 1 + 154 T + p^{3} T^{2} \)
79 \( 1 - 656 T + p^{3} T^{2} \)
83 \( 1 - 236 T + p^{3} T^{2} \)
89 \( 1 + 714 T + p^{3} T^{2} \)
97 \( 1 - 478 T + p^{3} T^{2} \)
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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

−9.374678929129614221412052914191, −8.856665068709210199922643360434, −7.62622683440126051190839342766, −6.67312837193659420704866554694, −5.98866124186556696428845030890, −5.06321636358969952852239906619, −4.07902915854288173432026733261, −2.78161930979285828500314655002, −1.35696926398637994243540140334, 0, 1.35696926398637994243540140334, 2.78161930979285828500314655002, 4.07902915854288173432026733261, 5.06321636358969952852239906619, 5.98866124186556696428845030890, 6.67312837193659420704866554694, 7.62622683440126051190839342766, 8.856665068709210199922643360434, 9.374678929129614221412052914191

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