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Properties

Label 2-15680-1.1-c1-0-28
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 − 11-s + 3·13-s + 15-s + 7·17-s + 4·19-s + 25-s + 5·27-s + 5·29-s + 10·31-s + 33-s + 4·37-s − 3·39-s + 10·41-s − 8·43-s + 2·45-s + 47-s − 7·51-s + 4·53-s + 55-s − 4·57-s − 10·61-s − 3·65-s + 12·67-s − 12·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.301·11-s + 0.832·13-s + 0.258·15-s + 1.69·17-s + 0.917·19-s + 1/5·25-s + 0.962·27-s + 0.928·29-s + 1.79·31-s + 0.174·33-s + 0.657·37-s − 0.480·39-s + 1.56·41-s − 1.21·43-s + 0.298·45-s + 0.145·47-s − 0.980·51-s + 0.549·53-s + 0.134·55-s − 0.529·57-s − 1.28·61-s − 0.372·65-s + 1.46·67-s − 1.42·71-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.798028095\)
\(L(\frac12)\) \(\approx\) \(1.798028095\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
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

−16.16517407886934, −15.52938470000691, −14.94612189776558, −14.19002881366323, −13.91906935662990, −13.23484884054183, −12.40854637964542, −12.02871648180258, −11.57607035443068, −11.02145254866583, −10.36900138761018, −9.883568938533115, −9.164061093246599, −8.278568783648006, −8.080252788500425, −7.354167513069710, −6.505021808954055, −5.979913647244092, −5.407118847751973, −4.786264999681162, −3.975055455048610, −3.112177190256358, −2.736615149502682, −1.243862540840742, −0.6994303937552847, 0.6994303937552847, 1.243862540840742, 2.736615149502682, 3.112177190256358, 3.975055455048610, 4.786264999681162, 5.407118847751973, 5.979913647244092, 6.505021808954055, 7.354167513069710, 8.080252788500425, 8.278568783648006, 9.164061093246599, 9.883568938533115, 10.36900138761018, 11.02145254866583, 11.57607035443068, 12.02871648180258, 12.40854637964542, 13.23484884054183, 13.91906935662990, 14.19002881366323, 14.94612189776558, 15.52938470000691, 16.16517407886934

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