Properties

Label 2-28e2-1.1-c1-0-12
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s + 4·13-s − 6·17-s + 2·19-s − 5·25-s + 4·27-s − 6·29-s − 4·31-s + 2·37-s − 8·39-s − 6·41-s − 8·43-s − 12·47-s + 12·51-s + 6·53-s − 4·57-s − 6·59-s − 8·61-s + 4·67-s − 2·73-s + 10·75-s − 8·79-s − 11·81-s − 6·83-s + 12·87-s + 6·89-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s + 1.10·13-s − 1.45·17-s + 0.458·19-s − 25-s + 0.769·27-s − 1.11·29-s − 0.718·31-s + 0.328·37-s − 1.28·39-s − 0.937·41-s − 1.21·43-s − 1.75·47-s + 1.68·51-s + 0.824·53-s − 0.529·57-s − 0.781·59-s − 1.02·61-s + 0.488·67-s − 0.234·73-s + 1.15·75-s − 0.900·79-s − 1.22·81-s − 0.658·83-s + 1.28·87-s + 0.635·89-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: \(1\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−10.02053664953109744946236323249, −9.044637307272642732451675861332, −8.199638847074308569053560326614, −7.01851678171449461328017406897, −6.24500065333065006116855945568, −5.53331414913354333592453982899, −4.55372588498345584946220609716, −3.42554077807776045337947490220, −1.73289286183865826039844070287, 0, 1.73289286183865826039844070287, 3.42554077807776045337947490220, 4.55372588498345584946220609716, 5.53331414913354333592453982899, 6.24500065333065006116855945568, 7.01851678171449461328017406897, 8.199638847074308569053560326614, 9.044637307272642732451675861332, 10.02053664953109744946236323249

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