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Properties

Label 2-15680-1.1-c1-0-47
Degree $2$
Conductor $15680$
Sign $-1$
Analytic cond. $125.205$
Root an. cond. $11.1895$
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-s − 5-s − 2·9-s − 11-s − 13-s + 15-s − 17-s + 4·23-s + 25-s + 5·27-s + 29-s − 10·31-s + 33-s + 8·37-s + 39-s − 2·41-s + 4·43-s + 2·45-s − 11·47-s + 51-s − 4·53-s + 55-s + 8·59-s + 10·61-s + 65-s + 4·67-s − 4·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.301·11-s − 0.277·13-s + 0.258·15-s − 0.242·17-s + 0.834·23-s + 1/5·25-s + 0.962·27-s + 0.185·29-s − 1.79·31-s + 0.174·33-s + 1.31·37-s + 0.160·39-s − 0.312·41-s + 0.609·43-s + 0.298·45-s − 1.60·47-s + 0.140·51-s − 0.549·53-s + 0.134·55-s + 1.04·59-s + 1.28·61-s + 0.124·65-s + 0.488·67-s − 0.481·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15680\)    =    \(2^{6} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(125.205\)
Root analytic conductor: \(11.1895\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 15680,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good3 \( 1 + T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−16.24558655058299, −15.89760135533770, −15.10703367858870, −14.55360519534031, −14.32202140253348, −13.26721984808080, −12.93074166555463, −12.37447315802090, −11.60689757978242, −11.26131236686708, −10.85518446314974, −10.12132066646338, −9.405524831193938, −8.866479778828952, −8.156436185934641, −7.667492778139169, −6.852171466487349, −6.431258846575529, −5.477917985353736, −5.201858107235287, −4.398950562913471, −3.593413821256091, −2.880374247690925, −2.087676771806778, −0.8920061365315337, 0, 0.8920061365315337, 2.087676771806778, 2.880374247690925, 3.593413821256091, 4.398950562913471, 5.201858107235287, 5.477917985353736, 6.431258846575529, 6.852171466487349, 7.667492778139169, 8.156436185934641, 8.866479778828952, 9.405524831193938, 10.12132066646338, 10.85518446314974, 11.26131236686708, 11.60689757978242, 12.37447315802090, 12.93074166555463, 13.26721984808080, 14.32202140253348, 14.55360519534031, 15.10703367858870, 15.89760135533770, 16.24558655058299

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