Properties

Label 2-15680-1.1-c1-0-19
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 $0$

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: \(0\)
Selberg data: \((2,\ 15680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.069612674\)
\(L(\frac12)\) \(\approx\) \(2.069612674\)
\(L(\frac{3}{2})\) 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

−16.08290016336319, −15.09899147909658, −14.93881838565298, −14.29606188758307, −14.01581061157693, −13.36464310502616, −12.39062196794915, −12.15884142327556, −11.53544565078424, −11.12853443318762, −10.20504752183517, −9.605202302033285, −9.057018491267000, −8.701716425527106, −7.860032745507402, −7.415626991539509, −6.723608877877547, −6.083643271468407, −5.324312731031795, −4.549157187434392, −3.807107230396754, −3.362188141204069, −2.460517415261194, −1.719631792978963, −0.5964336565036037, 0.5964336565036037, 1.719631792978963, 2.460517415261194, 3.362188141204069, 3.807107230396754, 4.549157187434392, 5.324312731031795, 6.083643271468407, 6.723608877877547, 7.415626991539509, 7.860032745507402, 8.701716425527106, 9.057018491267000, 9.605202302033285, 10.20504752183517, 11.12853443318762, 11.53544565078424, 12.15884142327556, 12.39062196794915, 13.36464310502616, 14.01581061157693, 14.29606188758307, 14.93881838565298, 15.09899147909658, 16.08290016336319

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