Properties

Degree 2
Conductor $ 2^{4} \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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3920} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3920,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.122110356$
$L(\frac12)$  $\approx$  $1.122110356$
$L(\frac{3}{2})$   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

−18.27030361154466, −17.66117402981086, −16.85270410770541, −16.15811208452532, −15.84287038433995, −15.09997267857505, −14.41892384388129, −13.82582081063167, −13.16582464628757, −12.54396295248069, −11.78992479054891, −11.07596235861219, −10.80680890880322, −9.824582644615483, −9.060844997255099, −8.454267864352546, −7.673758167649079, −7.282369155038000, −5.932965798793598, −5.712973805965996, −4.699487398975097, −3.797821477590716, −2.959711766926658, −2.142733537704904, −0.5831750063507187, 0.5831750063507187, 2.142733537704904, 2.959711766926658, 3.797821477590716, 4.699487398975097, 5.712973805965996, 5.932965798793598, 7.282369155038000, 7.673758167649079, 8.454267864352546, 9.060844997255099, 9.824582644615483, 10.80680890880322, 11.07596235861219, 11.78992479054891, 12.54396295248069, 13.16582464628757, 13.82582081063167, 14.41892384388129, 15.09997267857505, 15.84287038433995, 16.15811208452532, 16.85270410770541, 17.66117402981086, 18.27030361154466

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