Properties

Degree 2
Conductor $ 2^{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  − 8·3-s + 14·5-s − 208·7-s − 179·9-s − 536·11-s + 694·13-s − 112·15-s − 1.27e3·17-s + 1.11e3·19-s + 1.66e3·21-s + 3.21e3·23-s − 2.92e3·25-s + 3.37e3·27-s + 2.91e3·29-s − 2.62e3·31-s + 4.28e3·33-s − 2.91e3·35-s − 9.45e3·37-s − 5.55e3·39-s + 170·41-s − 1.99e4·43-s − 2.50e3·45-s + 32·47-s + 2.64e4·49-s + 1.02e4·51-s − 2.21e4·53-s − 7.50e3·55-s + ⋯
L(s)  = 1  − 0.513·3-s + 0.250·5-s − 1.60·7-s − 0.736·9-s − 1.33·11-s + 1.13·13-s − 0.128·15-s − 1.07·17-s + 0.706·19-s + 0.823·21-s + 1.26·23-s − 0.937·25-s + 0.891·27-s + 0.644·29-s − 0.490·31-s + 0.685·33-s − 0.401·35-s − 1.13·37-s − 0.584·39-s + 0.0157·41-s − 1.64·43-s − 0.184·45-s + 0.00211·47-s + 1.57·49-s + 0.550·51-s − 1.08·53-s − 0.334·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(32\)    =    \(2^{5}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(5\)
character  :  $\chi_{32} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 32,\ (\ :5/2),\ -1)\)
\(L(3)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 8 T + p^{5} T^{2} \)
5 \( 1 - 14 T + p^{5} T^{2} \)
7 \( 1 + 208 T + p^{5} T^{2} \)
11 \( 1 + 536 T + p^{5} T^{2} \)
13 \( 1 - 694 T + p^{5} T^{2} \)
17 \( 1 + 1278 T + p^{5} T^{2} \)
19 \( 1 - 1112 T + p^{5} T^{2} \)
23 \( 1 - 3216 T + p^{5} T^{2} \)
29 \( 1 - 2918 T + p^{5} T^{2} \)
31 \( 1 + 2624 T + p^{5} T^{2} \)
37 \( 1 + 9458 T + p^{5} T^{2} \)
41 \( 1 - 170 T + p^{5} T^{2} \)
43 \( 1 + 19928 T + p^{5} T^{2} \)
47 \( 1 - 32 T + p^{5} T^{2} \)
53 \( 1 + 22178 T + p^{5} T^{2} \)
59 \( 1 - 41480 T + p^{5} T^{2} \)
61 \( 1 - 15462 T + p^{5} T^{2} \)
67 \( 1 + 20744 T + p^{5} T^{2} \)
71 \( 1 - 28592 T + p^{5} T^{2} \)
73 \( 1 + 53670 T + p^{5} T^{2} \)
79 \( 1 + 69152 T + p^{5} T^{2} \)
83 \( 1 + 37800 T + p^{5} T^{2} \)
89 \( 1 + 126806 T + p^{5} T^{2} \)
97 \( 1 - 62290 T + p^{5} T^{2} \)
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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

−15.58207076215561775169825543226, −13.61336521132877859098282974396, −12.90187517906477616610015363559, −11.31857003666304005985439474032, −10.14003899981572732495857398518, −8.710965820746736160627417431972, −6.70672702287718643748508040768, −5.50818870005420743787505703747, −3.08685227262836591523781351559, 0, 3.08685227262836591523781351559, 5.50818870005420743787505703747, 6.70672702287718643748508040768, 8.710965820746736160627417431972, 10.14003899981572732495857398518, 11.31857003666304005985439474032, 12.90187517906477616610015363559, 13.61336521132877859098282974396, 15.58207076215561775169825543226

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