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  + 6.32·3-s − 25·5-s − 44.2·7-s − 203·9-s − 720.·11-s + 146·13-s − 158.·15-s − 702·17-s + 2.73e3·19-s − 280·21-s + 4.09e3·23-s + 625·25-s − 2.82e3·27-s + 4.01e3·29-s − 4.56e3·31-s − 4.56e3·33-s + 1.10e3·35-s + 1.47e4·37-s + 923.·39-s − 4.35e3·41-s + 1.24e4·43-s + 5.07e3·45-s − 6.01e3·47-s − 1.48e4·49-s − 4.43e3·51-s + 1.81e4·53-s + 1.80e4·55-s + ⋯
L(s)  = 1  + 0.405·3-s − 0.447·5-s − 0.341·7-s − 0.835·9-s − 1.79·11-s + 0.239·13-s − 0.181·15-s − 0.589·17-s + 1.73·19-s − 0.138·21-s + 1.61·23-s + 0.200·25-s − 0.744·27-s + 0.885·29-s − 0.853·31-s − 0.728·33-s + 0.152·35-s + 1.77·37-s + 0.0972·39-s − 0.404·41-s + 1.02·43-s + 0.373·45-s − 0.397·47-s − 0.883·49-s − 0.239·51-s + 0.887·53-s + 0.803·55-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  =  \((2,\ 320,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(1.509982416\)
\(L(\frac12)\)  \(\approx\)  \(1.509982416\)
\(L(\frac{7}{2})\)   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 \( 1 - 6.32T + 243T^{2} \)
7 \( 1 + 44.2T + 1.68e4T^{2} \)
11 \( 1 + 720.T + 1.61e5T^{2} \)
13 \( 1 - 146T + 3.71e5T^{2} \)
17 \( 1 + 702T + 1.41e6T^{2} \)
19 \( 1 - 2.73e3T + 2.47e6T^{2} \)
23 \( 1 - 4.09e3T + 6.43e6T^{2} \)
29 \( 1 - 4.01e3T + 2.05e7T^{2} \)
31 \( 1 + 4.56e3T + 2.86e7T^{2} \)
37 \( 1 - 1.47e4T + 6.93e7T^{2} \)
41 \( 1 + 4.35e3T + 1.15e8T^{2} \)
43 \( 1 - 1.24e4T + 1.47e8T^{2} \)
47 \( 1 + 6.01e3T + 2.29e8T^{2} \)
53 \( 1 - 1.81e4T + 4.18e8T^{2} \)
59 \( 1 + 1.97e4T + 7.14e8T^{2} \)
61 \( 1 - 4.21e4T + 8.44e8T^{2} \)
67 \( 1 - 1.61e4T + 1.35e9T^{2} \)
71 \( 1 + 4.54e4T + 1.80e9T^{2} \)
73 \( 1 - 2.62e4T + 2.07e9T^{2} \)
79 \( 1 - 8.67e3T + 3.07e9T^{2} \)
83 \( 1 + 9.87e4T + 3.93e9T^{2} \)
89 \( 1 - 3.05e4T + 5.58e9T^{2} \)
97 \( 1 - 6.68e4T + 8.58e9T^{2} \)
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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

−10.93311781962910836716675180448, −9.802471878091752943783771775280, −8.839626604553640661833342998693, −7.955254604777747824361998736997, −7.17746656229355761424452603713, −5.74475709357530976347433880492, −4.85878873764652386062272315053, −3.26337482456587039544030833630, −2.61585971034506651945096348300, −0.64567714477605399038840036874, 0.64567714477605399038840036874, 2.61585971034506651945096348300, 3.26337482456587039544030833630, 4.85878873764652386062272315053, 5.74475709357530976347433880492, 7.17746656229355761424452603713, 7.955254604777747824361998736997, 8.839626604553640661833342998693, 9.802471878091752943783771775280, 10.93311781962910836716675180448

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