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  − 6·3-s + 25·5-s − 118·7-s − 207·9-s − 192·11-s − 1.10e3·13-s − 150·15-s + 762·17-s + 2.74e3·19-s + 708·21-s + 1.56e3·23-s + 625·25-s + 2.70e3·27-s − 5.91e3·29-s − 6.86e3·31-s + 1.15e3·33-s − 2.95e3·35-s + 5.51e3·37-s + 6.63e3·39-s − 378·41-s + 2.43e3·43-s − 5.17e3·45-s + 1.31e4·47-s − 2.88e3·49-s − 4.57e3·51-s + 9.17e3·53-s − 4.80e3·55-s + ⋯
L(s)  = 1  − 0.384·3-s + 0.447·5-s − 0.910·7-s − 0.851·9-s − 0.478·11-s − 1.81·13-s − 0.172·15-s + 0.639·17-s + 1.74·19-s + 0.350·21-s + 0.617·23-s + 1/5·25-s + 0.712·27-s − 1.30·29-s − 1.28·31-s + 0.184·33-s − 0.407·35-s + 0.662·37-s + 0.698·39-s − 0.0351·41-s + 0.200·43-s − 0.380·45-s + 0.866·47-s − 0.171·49-s − 0.246·51-s + 0.448·53-s − 0.213·55-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  =  \((2,\ 320,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(1.085041162\)
\(L(\frac12)\)  \(\approx\)  \(1.085041162\)
\(L(\frac{7}{2})\)   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 \( 1 - p^{2} T \)
good3 \( 1 + 2 p T + p^{5} T^{2} \)
7 \( 1 + 118 T + p^{5} T^{2} \)
11 \( 1 + 192 T + p^{5} T^{2} \)
13 \( 1 + 1106 T + p^{5} T^{2} \)
17 \( 1 - 762 T + p^{5} T^{2} \)
19 \( 1 - 2740 T + p^{5} T^{2} \)
23 \( 1 - 1566 T + p^{5} T^{2} \)
29 \( 1 + 5910 T + p^{5} T^{2} \)
31 \( 1 + 6868 T + p^{5} T^{2} \)
37 \( 1 - 5518 T + p^{5} T^{2} \)
41 \( 1 + 378 T + p^{5} T^{2} \)
43 \( 1 - 2434 T + p^{5} T^{2} \)
47 \( 1 - 13122 T + p^{5} T^{2} \)
53 \( 1 - 9174 T + p^{5} T^{2} \)
59 \( 1 - 34980 T + p^{5} T^{2} \)
61 \( 1 - 9838 T + p^{5} T^{2} \)
67 \( 1 + 33722 T + p^{5} T^{2} \)
71 \( 1 - 70212 T + p^{5} T^{2} \)
73 \( 1 - 21986 T + p^{5} T^{2} \)
79 \( 1 - 4520 T + p^{5} T^{2} \)
83 \( 1 - 109074 T + p^{5} T^{2} \)
89 \( 1 - 38490 T + p^{5} T^{2} \)
97 \( 1 + 1918 T + p^{5} T^{2} \)
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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

−10.77123803035028232960763292825, −9.692436294261665437103490491608, −9.313143357024145647337940734346, −7.74076691050945123801529012318, −6.97379482944925067542154875019, −5.62036463732788811931789966212, −5.19754526426276643341124078864, −3.36475918352649418176800414997, −2.41797532320575322759885164950, −0.56142460750480993701946855652, 0.56142460750480993701946855652, 2.41797532320575322759885164950, 3.36475918352649418176800414997, 5.19754526426276643341124078864, 5.62036463732788811931789966212, 6.97379482944925067542154875019, 7.74076691050945123801529012318, 9.313143357024145647337940734346, 9.692436294261665437103490491608, 10.77123803035028232960763292825

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