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  + 13.4·3-s + 25·5-s − 138.·7-s − 62.9·9-s + 259.·11-s − 154·13-s + 335.·15-s + 178·17-s + 965.·19-s − 1.86e3·21-s − 2.63e3·23-s + 625·25-s − 4.10e3·27-s − 4.11e3·29-s − 3.15e3·31-s + 3.48e3·33-s − 3.46e3·35-s − 7.44e3·37-s − 2.06e3·39-s + 7.27e3·41-s − 1.79e4·43-s − 1.57e3·45-s + 7.41e3·47-s + 2.41e3·49-s + 2.38e3·51-s − 3.22e4·53-s + 6.48e3·55-s + ⋯
L(s)  = 1  + 0.860·3-s + 0.447·5-s − 1.06·7-s − 0.259·9-s + 0.646·11-s − 0.252·13-s + 0.384·15-s + 0.149·17-s + 0.613·19-s − 0.920·21-s − 1.03·23-s + 0.200·25-s − 1.08·27-s − 0.907·29-s − 0.590·31-s + 0.556·33-s − 0.478·35-s − 0.893·37-s − 0.217·39-s + 0.675·41-s − 1.47·43-s − 0.115·45-s + 0.489·47-s + 0.143·49-s + 0.128·51-s − 1.57·53-s + 0.289·55-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 \( 1 - 25T \)
good3 \( 1 - 13.4T + 243T^{2} \)
7 \( 1 + 138.T + 1.68e4T^{2} \)
11 \( 1 - 259.T + 1.61e5T^{2} \)
13 \( 1 + 154T + 3.71e5T^{2} \)
17 \( 1 - 178T + 1.41e6T^{2} \)
19 \( 1 - 965.T + 2.47e6T^{2} \)
23 \( 1 + 2.63e3T + 6.43e6T^{2} \)
29 \( 1 + 4.11e3T + 2.05e7T^{2} \)
31 \( 1 + 3.15e3T + 2.86e7T^{2} \)
37 \( 1 + 7.44e3T + 6.93e7T^{2} \)
41 \( 1 - 7.27e3T + 1.15e8T^{2} \)
43 \( 1 + 1.79e4T + 1.47e8T^{2} \)
47 \( 1 - 7.41e3T + 2.29e8T^{2} \)
53 \( 1 + 3.22e4T + 4.18e8T^{2} \)
59 \( 1 - 3.40e4T + 7.14e8T^{2} \)
61 \( 1 + 2.67e4T + 8.44e8T^{2} \)
67 \( 1 - 4.98e4T + 1.35e9T^{2} \)
71 \( 1 + 5.41e4T + 1.80e9T^{2} \)
73 \( 1 + 1.85e4T + 2.07e9T^{2} \)
79 \( 1 - 8.67e4T + 3.07e9T^{2} \)
83 \( 1 + 7.86e4T + 3.93e9T^{2} \)
89 \( 1 + 1.07e5T + 5.58e9T^{2} \)
97 \( 1 + 1.08e5T + 8.58e9T^{2} \)
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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

−9.945554062805176147158495009041, −9.458577640981746493848796912294, −8.600973770804502783041058379535, −7.49448363311508899974509643248, −6.43465355029748116204503022177, −5.47640448027724189083629960911, −3.82779672076084275475475817841, −3.00121562687584385147275153389, −1.78668976466563719064047722148, 0, 1.78668976466563719064047722148, 3.00121562687584385147275153389, 3.82779672076084275475475817841, 5.47640448027724189083629960911, 6.43465355029748116204503022177, 7.49448363311508899974509643248, 8.600973770804502783041058379535, 9.458577640981746493848796912294, 9.945554062805176147158495009041

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