Properties

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

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  =  \(1\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ -1)\)
\(L(3)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{7}{2})\)   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

−10.61578830480697537102032526218, −9.030032925809033336983419967336, −8.527347101144338499907190709923, −7.61228413897650899581040669800, −6.50534313788166096068015292680, −5.09683710516651555382329038585, −4.34516645683517783212093349085, −2.85172142027739288391881065617, −1.64116969739870344240112165873, 0, 1.64116969739870344240112165873, 2.85172142027739288391881065617, 4.34516645683517783212093349085, 5.09683710516651555382329038585, 6.50534313788166096068015292680, 7.61228413897650899581040669800, 8.527347101144338499907190709923, 9.030032925809033336983419967336, 10.61578830480697537102032526218

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