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  − 4·3-s − 25·5-s − 192·7-s − 227·9-s − 148·11-s − 286·13-s + 100·15-s − 1.67e3·17-s + 1.06e3·19-s + 768·21-s − 2.97e3·23-s + 625·25-s + 1.88e3·27-s + 3.41e3·29-s + 2.44e3·31-s + 592·33-s + 4.80e3·35-s − 182·37-s + 1.14e3·39-s − 9.39e3·41-s − 1.24e3·43-s + 5.67e3·45-s + 1.20e4·47-s + 2.00e4·49-s + 6.71e3·51-s − 2.38e4·53-s + 3.70e3·55-s + ⋯
L(s)  = 1  − 0.256·3-s − 0.447·5-s − 1.48·7-s − 0.934·9-s − 0.368·11-s − 0.469·13-s + 0.114·15-s − 1.40·17-s + 0.673·19-s + 0.380·21-s − 1.17·23-s + 1/5·25-s + 0.496·27-s + 0.752·29-s + 0.457·31-s + 0.0946·33-s + 0.662·35-s − 0.0218·37-s + 0.120·39-s − 0.873·41-s − 0.102·43-s + 0.417·45-s + 0.798·47-s + 1.19·49-s + 0.361·51-s − 1.16·53-s + 0.164·55-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  =  \((2,\ 320,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(0.4335863759\)
\(L(\frac12)\)  \(\approx\)  \(0.4335863759\)
\(L(\frac{7}{2})\)   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 \( 1 + 4 T + p^{5} T^{2} \)
7 \( 1 + 192 T + p^{5} T^{2} \)
11 \( 1 + 148 T + p^{5} T^{2} \)
13 \( 1 + 22 p T + p^{5} T^{2} \)
17 \( 1 + 1678 T + p^{5} T^{2} \)
19 \( 1 - 1060 T + p^{5} T^{2} \)
23 \( 1 + 2976 T + p^{5} T^{2} \)
29 \( 1 - 3410 T + p^{5} T^{2} \)
31 \( 1 - 2448 T + p^{5} T^{2} \)
37 \( 1 + 182 T + p^{5} T^{2} \)
41 \( 1 + 9398 T + p^{5} T^{2} \)
43 \( 1 + 1244 T + p^{5} T^{2} \)
47 \( 1 - 12088 T + p^{5} T^{2} \)
53 \( 1 + 23846 T + p^{5} T^{2} \)
59 \( 1 + 20020 T + p^{5} T^{2} \)
61 \( 1 + 32302 T + p^{5} T^{2} \)
67 \( 1 - 60972 T + p^{5} T^{2} \)
71 \( 1 - 32648 T + p^{5} T^{2} \)
73 \( 1 + 38774 T + p^{5} T^{2} \)
79 \( 1 - 33360 T + p^{5} T^{2} \)
83 \( 1 - 16716 T + p^{5} T^{2} \)
89 \( 1 - 101370 T + p^{5} T^{2} \)
97 \( 1 + 119038 T + p^{5} T^{2} \)
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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

−10.80190430644786197843231656721, −9.876535428533976510117621177181, −8.983025923280385841352666233353, −7.962636256946231908485602253641, −6.75817530437946425666347344241, −6.04759242787700065344314799339, −4.78919962335594940297119523027, −3.45901557683531458464719008344, −2.49611108850417448478443315898, −0.34384914638445958329004326917, 0.34384914638445958329004326917, 2.49611108850417448478443315898, 3.45901557683531458464719008344, 4.78919962335594940297119523027, 6.04759242787700065344314799339, 6.75817530437946425666347344241, 7.962636256946231908485602253641, 8.983025923280385841352666233353, 9.876535428533976510117621177181, 10.80190430644786197843231656721

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