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Properties

Label 2-28e2-1.1-c1-0-15
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  − 3·9-s − 4·11-s − 8·23-s − 5·25-s + 2·29-s − 6·37-s + 12·43-s − 10·53-s − 4·67-s − 16·71-s − 8·79-s + 9·81-s + 12·99-s + 20·107-s + 18·109-s + 2·113-s + ⋯
L(s)  = 1  − 9-s − 1.20·11-s − 1.66·23-s − 25-s + 0.371·29-s − 0.986·37-s + 1.82·43-s − 1.37·53-s − 0.488·67-s − 1.89·71-s − 0.900·79-s + 81-s + 1.20·99-s + 1.93·107-s + 1.72·109-s + 0.188·113-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: Trivial
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 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−9.986083473283480406143801341179, −8.949658951219617610034987207596, −8.102431901520003899942818252728, −7.49516008990432878175521888778, −6.11693101856965229843525121209, −5.56423915340359066406328786043, −4.41080963086329800320550064306, −3.16972888563709012933599163579, −2.12708221183439266543513013448, 0, 2.12708221183439266543513013448, 3.16972888563709012933599163579, 4.41080963086329800320550064306, 5.56423915340359066406328786043, 6.11693101856965229843525121209, 7.49516008990432878175521888778, 8.102431901520003899942818252728, 8.949658951219617610034987207596, 9.986083473283480406143801341179

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