Properties

Label 2-28e2-1.1-c1-0-11
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4·5-s + 9-s + 8·15-s + 2·17-s − 2·19-s − 8·23-s + 11·25-s − 4·27-s + 2·29-s + 4·31-s − 6·37-s + 2·41-s − 8·43-s + 4·45-s − 4·47-s + 4·51-s − 10·53-s − 4·57-s + 6·59-s − 4·61-s + 12·67-s − 16·69-s + 14·73-s + 22·75-s + 8·79-s − 11·81-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 1/3·9-s + 2.06·15-s + 0.485·17-s − 0.458·19-s − 1.66·23-s + 11/5·25-s − 0.769·27-s + 0.371·29-s + 0.718·31-s − 0.986·37-s + 0.312·41-s − 1.21·43-s + 0.596·45-s − 0.583·47-s + 0.560·51-s − 1.37·53-s − 0.529·57-s + 0.781·59-s − 0.512·61-s + 1.46·67-s − 1.92·69-s + 1.63·73-s + 2.54·75-s + 0.900·79-s − 1.22·81-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: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.866540363\)
\(L(\frac12)\) \(\approx\) \(2.866540363\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−9.835763078329112662938551691511, −9.676028605922657643630885041741, −8.589598276995096865608741769842, −8.037894346653340821155380595819, −6.69299348503088401254902880103, −5.96146004102990052123544748972, −4.99083605129769608898331014735, −3.56791835941750047835108384805, −2.48337724162238959661867349800, −1.73163501201493931998851829334, 1.73163501201493931998851829334, 2.48337724162238959661867349800, 3.56791835941750047835108384805, 4.99083605129769608898331014735, 5.96146004102990052123544748972, 6.69299348503088401254902880103, 8.037894346653340821155380595819, 8.589598276995096865608741769842, 9.676028605922657643630885041741, 9.835763078329112662938551691511

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