Properties

Label 2-15680-1.1-c1-0-45
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 − 5·13-s + 15-s − 17-s + 6·19-s + 4·23-s + 25-s + 5·27-s − 3·29-s + 2·31-s + 33-s − 8·37-s + 5·39-s + 10·41-s − 2·43-s + 2·45-s − 7·47-s + 51-s + 2·53-s + 55-s − 6·57-s − 14·59-s − 8·61-s + 5·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.301·11-s − 1.38·13-s + 0.258·15-s − 0.242·17-s + 1.37·19-s + 0.834·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s + 0.359·31-s + 0.174·33-s − 1.31·37-s + 0.800·39-s + 1.56·41-s − 0.304·43-s + 0.298·45-s − 1.02·47-s + 0.140·51-s + 0.274·53-s + 0.134·55-s − 0.794·57-s − 1.82·59-s − 1.02·61-s + 0.620·65-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: $\chi_{15680} (1, \cdot )$
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 \( 1 + 5 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−16.39815852364548, −15.63236137186863, −15.30712590257305, −14.58691385386296, −14.09215870036497, −13.59291306198741, −12.66389091779572, −12.34979329160755, −11.80074174429637, −11.18912663214094, −10.85210135548439, −10.04947918902059, −9.440882746432201, −8.953625736949270, −8.070965832766780, −7.603207621690220, −7.013806294667084, −6.358878259478575, −5.508398600762048, −5.059451891579861, −4.580930097484760, −3.450242881247549, −2.946288340719025, −2.098837857660366, −0.8733868477528976, 0, 0.8733868477528976, 2.098837857660366, 2.946288340719025, 3.450242881247549, 4.580930097484760, 5.059451891579861, 5.508398600762048, 6.358878259478575, 7.013806294667084, 7.603207621690220, 8.070965832766780, 8.953625736949270, 9.440882746432201, 10.04947918902059, 10.85210135548439, 11.18912663214094, 11.80074174429637, 12.34979329160755, 12.66389091779572, 13.59291306198741, 14.09215870036497, 14.58691385386296, 15.30712590257305, 15.63236137186863, 16.39815852364548

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