Properties

Degree 2
Conductor $ 2^{3} \cdot 5 \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 3·9-s + 4·11-s + 2·13-s − 2·17-s − 4·19-s + 4·23-s + 25-s − 2·29-s + 8·31-s + 6·37-s + 6·41-s − 8·43-s + 3·45-s − 4·47-s + 6·53-s − 4·55-s + 4·59-s + 2·61-s − 2·65-s + 8·67-s + 6·73-s + 9·81-s + 16·83-s + 2·85-s + 6·89-s + 4·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 9-s + 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.986·37-s + 0.937·41-s − 1.21·43-s + 0.447·45-s − 0.583·47-s + 0.824·53-s − 0.539·55-s + 0.520·59-s + 0.256·61-s − 0.248·65-s + 0.977·67-s + 0.702·73-s + 81-s + 1.75·83-s + 0.216·85-s + 0.635·89-s + 0.410·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1960} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1960,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.526149348$
$L(\frac12)$  $\approx$  $1.526149348$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.45730110487526, −19.00425740005397, −18.06973844285318, −17.40620342418872, −16.84564473003505, −16.34093359101843, −15.34395734221079, −14.87906767786108, −14.25778691959030, −13.48423237410861, −12.81039247567550, −11.88725533070116, −11.40206280806296, −10.88531981976035, −9.863905830132217, −8.938571023394552, −8.568492662536041, −7.731489384092577, −6.564070512491315, −6.277880394975758, −5.085169381872179, −4.179400139539493, −3.383016163825964, −2.292151368867222, −0.8341085067229485, 0.8341085067229485, 2.292151368867222, 3.383016163825964, 4.179400139539493, 5.085169381872179, 6.277880394975758, 6.564070512491315, 7.731489384092577, 8.568492662536041, 8.938571023394552, 9.863905830132217, 10.88531981976035, 11.40206280806296, 11.88725533070116, 12.81039247567550, 13.48423237410861, 14.25778691959030, 14.87906767786108, 15.34395734221079, 16.34093359101843, 16.84564473003505, 17.40620342418872, 18.06973844285318, 19.00425740005397, 19.45730110487526

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