Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 18.4·3-s + 25·5-s + 121.·7-s + 98.1·9-s + 438.·11-s + 758.·13-s + 461.·15-s − 1.53e3·17-s − 75.8·19-s + 2.25e3·21-s + 3.69e3·23-s + 625·25-s − 2.67e3·27-s − 6.32e3·29-s + 2.69e3·31-s + 8.09e3·33-s + 3.04e3·35-s + 7.25e3·37-s + 1.40e4·39-s + 4.91e3·41-s − 2.53e3·43-s + 2.45e3·45-s + 1.13e4·47-s − 1.94e3·49-s − 2.83e4·51-s − 2.94e4·53-s + 1.09e4·55-s + ⋯
L(s)  = 1  + 1.18·3-s + 0.447·5-s + 0.940·7-s + 0.404·9-s + 1.09·11-s + 1.24·13-s + 0.529·15-s − 1.28·17-s − 0.0482·19-s + 1.11·21-s + 1.45·23-s + 0.200·25-s − 0.706·27-s − 1.39·29-s + 0.502·31-s + 1.29·33-s + 0.420·35-s + 0.870·37-s + 1.47·39-s + 0.456·41-s − 0.208·43-s + 0.180·45-s + 0.751·47-s − 0.115·49-s − 1.52·51-s − 1.43·53-s + 0.488·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{320} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(4.319842565\)
\(L(\frac12)\)  \(\approx\)  \(4.319842565\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - 25T \)
good3 \( 1 - 18.4T + 243T^{2} \)
7 \( 1 - 121.T + 1.68e4T^{2} \)
11 \( 1 - 438.T + 1.61e5T^{2} \)
13 \( 1 - 758.T + 3.71e5T^{2} \)
17 \( 1 + 1.53e3T + 1.41e6T^{2} \)
19 \( 1 + 75.8T + 2.47e6T^{2} \)
23 \( 1 - 3.69e3T + 6.43e6T^{2} \)
29 \( 1 + 6.32e3T + 2.05e7T^{2} \)
31 \( 1 - 2.69e3T + 2.86e7T^{2} \)
37 \( 1 - 7.25e3T + 6.93e7T^{2} \)
41 \( 1 - 4.91e3T + 1.15e8T^{2} \)
43 \( 1 + 2.53e3T + 1.47e8T^{2} \)
47 \( 1 - 1.13e4T + 2.29e8T^{2} \)
53 \( 1 + 2.94e4T + 4.18e8T^{2} \)
59 \( 1 - 5.68e3T + 7.14e8T^{2} \)
61 \( 1 - 4.80e4T + 8.44e8T^{2} \)
67 \( 1 + 3.95e4T + 1.35e9T^{2} \)
71 \( 1 - 1.26e4T + 1.80e9T^{2} \)
73 \( 1 - 5.79e4T + 2.07e9T^{2} \)
79 \( 1 + 2.95e4T + 3.07e9T^{2} \)
83 \( 1 + 1.12e5T + 3.93e9T^{2} \)
89 \( 1 - 6.69e4T + 5.58e9T^{2} \)
97 \( 1 - 1.31e5T + 8.58e9T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.03067228187167203717958131934, −9.469964174139763375517953156896, −8.890103223737571750570377065123, −8.237197059493658494651205127485, −7.05268493215824338189375202288, −5.96069053823299166018170098387, −4.54108550092915366024449072680, −3.51017760547060839648702056210, −2.21950048554709221273862364784, −1.23730372781136489716802478269, 1.23730372781136489716802478269, 2.21950048554709221273862364784, 3.51017760547060839648702056210, 4.54108550092915366024449072680, 5.96069053823299166018170098387, 7.05268493215824338189375202288, 8.237197059493658494651205127485, 8.890103223737571750570377065123, 9.469964174139763375517953156896, 11.03067228187167203717958131934

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