Properties

Label 2-28e2-1.1-c3-0-43
Degree $2$
Conductor $784$
Sign $-1$
Analytic cond. $46.2574$
Root an. cond. $6.80128$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.07·3-s + 19.7·5-s + 23.0·9-s + 14·11-s − 50.9·13-s − 140·15-s − 1.41·17-s − 1.41·19-s − 140·23-s + 267·25-s + 28.2·27-s − 286·29-s − 93.3·31-s − 98.9·33-s − 38·37-s + 360·39-s + 125.·41-s + 34·43-s + 455.·45-s + 523.·47-s + 10.0·51-s − 74·53-s + 277.·55-s + 10.0·57-s + 434.·59-s − 14.1·61-s − 1.00e3·65-s + ⋯
L(s)  = 1  − 1.36·3-s + 1.77·5-s + 0.851·9-s + 0.383·11-s − 1.08·13-s − 2.40·15-s − 0.0201·17-s − 0.0170·19-s − 1.26·23-s + 2.13·25-s + 0.201·27-s − 1.83·29-s − 0.540·31-s − 0.522·33-s − 0.168·37-s + 1.47·39-s + 0.479·41-s + 0.120·43-s + 1.50·45-s + 1.62·47-s + 0.0274·51-s − 0.191·53-s + 0.679·55-s + 0.0232·57-s + 0.958·59-s − 0.0296·61-s − 1.92·65-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: $-1$
Analytic conductor: \(46.2574\)
Root analytic conductor: \(6.80128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
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 \( 1 + 7.07T + 27T^{2} \)
5 \( 1 - 19.7T + 125T^{2} \)
11 \( 1 - 14T + 1.33e3T^{2} \)
13 \( 1 + 50.9T + 2.19e3T^{2} \)
17 \( 1 + 1.41T + 4.91e3T^{2} \)
19 \( 1 + 1.41T + 6.85e3T^{2} \)
23 \( 1 + 140T + 1.21e4T^{2} \)
29 \( 1 + 286T + 2.43e4T^{2} \)
31 \( 1 + 93.3T + 2.97e4T^{2} \)
37 \( 1 + 38T + 5.06e4T^{2} \)
41 \( 1 - 125.T + 6.89e4T^{2} \)
43 \( 1 - 34T + 7.95e4T^{2} \)
47 \( 1 - 523.T + 1.03e5T^{2} \)
53 \( 1 + 74T + 1.48e5T^{2} \)
59 \( 1 - 434.T + 2.05e5T^{2} \)
61 \( 1 + 14.1T + 2.26e5T^{2} \)
67 \( 1 + 684T + 3.00e5T^{2} \)
71 \( 1 + 588T + 3.57e5T^{2} \)
73 \( 1 - 270.T + 3.89e5T^{2} \)
79 \( 1 + 1.22e3T + 4.93e5T^{2} \)
83 \( 1 - 422.T + 5.71e5T^{2} \)
89 \( 1 + 618.T + 7.04e5T^{2} \)
97 \( 1 + 1.48e3T + 9.12e5T^{2} \)
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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

−9.684022984397692245454261874020, −8.968678444054366104727739920644, −7.43646476898576017409533630218, −6.58593248224015170219759449413, −5.67138315321316035460112516126, −5.48367601410815890892870391696, −4.25213160156057497604573999231, −2.47013692941292064025370315172, −1.48133480202169202166090406584, 0, 1.48133480202169202166090406584, 2.47013692941292064025370315172, 4.25213160156057497604573999231, 5.48367601410815890892870391696, 5.67138315321316035460112516126, 6.58593248224015170219759449413, 7.43646476898576017409533630218, 8.968678444054366104727739920644, 9.684022984397692245454261874020

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