Properties

Degree 2
Conductor $ 2^{2} \cdot 5 \cdot 11 \cdot 17^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 5-s + 4·7-s + 9-s + 11-s − 4·13-s − 2·15-s − 4·19-s + 8·21-s + 6·23-s + 25-s − 4·27-s + 6·29-s − 8·31-s + 2·33-s − 4·35-s − 2·37-s − 8·39-s − 6·41-s + 8·43-s − 45-s + 6·47-s + 9·49-s − 6·53-s − 55-s − 8·57-s − 12·59-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.516·15-s − 0.917·19-s + 1.74·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s − 1.43·31-s + 0.348·33-s − 0.676·35-s − 0.328·37-s − 1.28·39-s − 0.937·41-s + 1.21·43-s − 0.149·45-s + 0.875·47-s + 9/7·49-s − 0.824·53-s − 0.134·55-s − 1.05·57-s − 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63580 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(63580\)    =    \(2^{2} \cdot 5 \cdot 11 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{63580} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 63580,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
11 \( 1 - T \)
17 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.38337029318576, −14.20843858173657, −13.75525316608578, −12.98794331697894, −12.42583675595913, −12.11628521932130, −11.38871509610595, −10.91321038355804, −10.62862538467505, −9.763859358424710, −9.161041250218935, −8.819543726734902, −8.346293884278367, −7.795668124864466, −7.452898080229219, −6.929129119449673, −6.150959732786326, −5.259415282097052, −4.875546303834769, −4.326045597516753, −3.719526407559753, −3.000739907932986, −2.413763785576306, −1.860871914032008, −1.144506835938141, 0, 1.144506835938141, 1.860871914032008, 2.413763785576306, 3.000739907932986, 3.719526407559753, 4.326045597516753, 4.875546303834769, 5.259415282097052, 6.150959732786326, 6.929129119449673, 7.452898080229219, 7.795668124864466, 8.346293884278367, 8.819543726734902, 9.161041250218935, 9.763859358424710, 10.62862538467505, 10.91321038355804, 11.38871509610595, 12.11628521932130, 12.42583675595913, 12.98794331697894, 13.75525316608578, 14.20843858173657, 14.38337029318576

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