Properties

Label 2-28e2-1.1-c3-0-47
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

−9.128411349068611003610056321677, −8.527507421274420929489114186294, −7.88210339607289002893571701690, −7.30859896895116771811050638705, −6.07977814132360318667575791600, −4.46276853386934438296346152520, −3.71063663694475051529453530851, −3.17908902205713486983845163468, −1.65669549311643921673767257976, 0, 1.65669549311643921673767257976, 3.17908902205713486983845163468, 3.71063663694475051529453530851, 4.46276853386934438296346152520, 6.07977814132360318667575791600, 7.30859896895116771811050638705, 7.88210339607289002893571701690, 8.527507421274420929489114186294, 9.128411349068611003610056321677

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