Properties

Label 2-15680-1.1-c1-0-46
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 5-s + 6·9-s − 11-s + 3·13-s − 3·15-s + 3·17-s + 6·19-s − 4·23-s + 25-s + 9·27-s + 29-s − 6·31-s − 3·33-s + 9·39-s − 6·41-s + 6·43-s − 6·45-s + 9·47-s + 9·51-s + 10·53-s + 55-s + 18·57-s − 6·59-s − 3·65-s + 14·67-s − 12·69-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.447·5-s + 2·9-s − 0.301·11-s + 0.832·13-s − 0.774·15-s + 0.727·17-s + 1.37·19-s − 0.834·23-s + 1/5·25-s + 1.73·27-s + 0.185·29-s − 1.07·31-s − 0.522·33-s + 1.44·39-s − 0.937·41-s + 0.914·43-s − 0.894·45-s + 1.31·47-s + 1.26·51-s + 1.37·53-s + 0.134·55-s + 2.38·57-s − 0.781·59-s − 0.372·65-s + 1.71·67-s − 1.44·69-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: $\chi_{15680} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.630746506\)
\(L(\frac12)\) \(\approx\) \(4.630746506\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.85740380983176, −15.43256208582063, −14.83531953630965, −14.24669266810947, −13.86479800417941, −13.43903241322711, −12.76994394864921, −12.18729571542309, −11.60069630531167, −10.78359849497468, −10.17704362491126, −9.630267574047913, −9.017783882072363, −8.541090643923492, −7.958823188871963, −7.455215319146138, −7.064870278297244, −5.967609846901284, −5.336152557924030, −4.361057484822262, −3.687987707237353, −3.335850026828528, −2.572466633011415, −1.763228942015554, −0.8832272889440027, 0.8832272889440027, 1.763228942015554, 2.572466633011415, 3.335850026828528, 3.687987707237353, 4.361057484822262, 5.336152557924030, 5.967609846901284, 7.064870278297244, 7.455215319146138, 7.958823188871963, 8.541090643923492, 9.017783882072363, 9.630267574047913, 10.17704362491126, 10.78359849497468, 11.60069630531167, 12.18729571542309, 12.76994394864921, 13.43903241322711, 13.86479800417941, 14.24669266810947, 14.83531953630965, 15.43256208582063, 15.85740380983176

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