Properties

Label 2-28e2-1.1-c3-0-46
Degree $2$
Conductor $784$
Sign $-1$
Analytic cond. $46.2574$
Root an. cond. $6.80128$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7·5-s − 26·9-s − 35·11-s + 66·13-s + 7·15-s + 59·17-s − 137·19-s + 7·23-s − 76·25-s − 53·27-s + 106·29-s − 75·31-s − 35·33-s + 11·37-s + 66·39-s − 498·41-s − 260·43-s − 182·45-s + 171·47-s + 59·51-s − 417·53-s − 245·55-s − 137·57-s + 17·59-s + 51·61-s + 462·65-s + ⋯
L(s)  = 1  + 0.192·3-s + 0.626·5-s − 0.962·9-s − 0.959·11-s + 1.40·13-s + 0.120·15-s + 0.841·17-s − 1.65·19-s + 0.0634·23-s − 0.607·25-s − 0.377·27-s + 0.678·29-s − 0.434·31-s − 0.184·33-s + 0.0488·37-s + 0.270·39-s − 1.89·41-s − 0.922·43-s − 0.602·45-s + 0.530·47-s + 0.161·51-s − 1.08·53-s − 0.600·55-s − 0.318·57-s + 0.0375·59-s + 0.107·61-s + 0.881·65-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: yes
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 \( 1 - T + p^{3} T^{2} \)
5 \( 1 - 7 T + p^{3} T^{2} \)
11 \( 1 + 35 T + p^{3} T^{2} \)
13 \( 1 - 66 T + p^{3} T^{2} \)
17 \( 1 - 59 T + p^{3} T^{2} \)
19 \( 1 + 137 T + p^{3} T^{2} \)
23 \( 1 - 7 T + p^{3} T^{2} \)
29 \( 1 - 106 T + p^{3} T^{2} \)
31 \( 1 + 75 T + p^{3} T^{2} \)
37 \( 1 - 11 T + p^{3} T^{2} \)
41 \( 1 + 498 T + p^{3} T^{2} \)
43 \( 1 + 260 T + p^{3} T^{2} \)
47 \( 1 - 171 T + p^{3} T^{2} \)
53 \( 1 + 417 T + p^{3} T^{2} \)
59 \( 1 - 17 T + p^{3} T^{2} \)
61 \( 1 - 51 T + p^{3} T^{2} \)
67 \( 1 + 439 T + p^{3} T^{2} \)
71 \( 1 - 784 T + p^{3} T^{2} \)
73 \( 1 - 295 T + p^{3} T^{2} \)
79 \( 1 - 495 T + p^{3} T^{2} \)
83 \( 1 + 932 T + p^{3} T^{2} \)
89 \( 1 + 873 T + p^{3} T^{2} \)
97 \( 1 + 290 T + p^{3} T^{2} \)
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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

−9.467028984163928199362895813576, −8.394044462086092177065674788532, −8.165974958671036572500893817969, −6.66982069346941180604386068820, −5.91251067461905381501729832388, −5.17007414542838741455380662210, −3.78499243702856244046725272724, −2.77427379711381541627513026585, −1.65609860242055915501292482839, 0, 1.65609860242055915501292482839, 2.77427379711381541627513026585, 3.78499243702856244046725272724, 5.17007414542838741455380662210, 5.91251067461905381501729832388, 6.66982069346941180604386068820, 8.165974958671036572500893817969, 8.394044462086092177065674788532, 9.467028984163928199362895813576

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