Properties

Label 2-15680-1.1-c1-0-34
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 − 6·11-s − 4·13-s + 15-s − 2·19-s + 3·23-s + 25-s + 5·27-s + 3·29-s + 8·31-s + 6·33-s + 4·37-s + 4·39-s − 9·41-s − 7·43-s + 2·45-s + 6·53-s + 6·55-s + 2·57-s + 6·59-s + 5·61-s + 4·65-s + 5·67-s − 3·69-s + 6·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2/3·9-s − 1.80·11-s − 1.10·13-s + 0.258·15-s − 0.458·19-s + 0.625·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s + 1.43·31-s + 1.04·33-s + 0.657·37-s + 0.640·39-s − 1.40·41-s − 1.06·43-s + 0.298·45-s + 0.824·53-s + 0.809·55-s + 0.264·57-s + 0.781·59-s + 0.640·61-s + 0.496·65-s + 0.610·67-s − 0.361·69-s + 0.712·71-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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−16.28095127247319, −15.74385747239284, −15.10829335126620, −14.86330306517881, −13.99928301339455, −13.41957455731746, −12.90425772712353, −12.23142759998565, −11.86052666608810, −11.20836920337390, −10.66692669750337, −10.13123501709273, −9.669833457571787, −8.545489765634843, −8.286905486974323, −7.685932782399964, −6.888302390408729, −6.428608900179182, −5.443643437919346, −5.068223308113141, −4.627576720400164, −3.517338518318765, −2.701900821096220, −2.320129580860657, −0.7784328724766832, 0, 0.7784328724766832, 2.320129580860657, 2.701900821096220, 3.517338518318765, 4.627576720400164, 5.068223308113141, 5.443643437919346, 6.428608900179182, 6.888302390408729, 7.685932782399964, 8.286905486974323, 8.545489765634843, 9.669833457571787, 10.13123501709273, 10.66692669750337, 11.20836920337390, 11.86052666608810, 12.23142759998565, 12.90425772712353, 13.41957455731746, 13.99928301339455, 14.86330306517881, 15.10829335126620, 15.74385747239284, 16.28095127247319

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