Properties

Degree 2
Conductor $ 2^{3} \cdot 5 \cdot 29^{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  + 5-s − 4·7-s − 3·9-s − 4·11-s − 2·13-s − 2·17-s − 4·19-s + 4·23-s + 25-s + 8·31-s − 4·35-s − 6·37-s + 6·41-s + 8·43-s − 3·45-s − 4·47-s + 9·49-s + 6·53-s − 4·55-s − 4·59-s + 2·61-s + 12·63-s − 2·65-s + 8·67-s + 6·73-s + 16·77-s + 9·81-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-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 + 1.43·31-s − 0.676·35-s − 0.986·37-s + 0.937·41-s + 1.21·43-s − 0.447·45-s − 0.583·47-s + 9/7·49-s + 0.824·53-s − 0.539·55-s − 0.520·59-s + 0.256·61-s + 1.51·63-s − 0.248·65-s + 0.977·67-s + 0.702·73-s + 1.82·77-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(33640\)    =    \(2^{3} \cdot 5 \cdot 29^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{33640} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 33640,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;29\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
29 \( 1 \)
good3 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
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

−15.37549537310899, −14.74048752760684, −14.12860792490237, −13.60331584853034, −13.15571969785026, −12.62316671210797, −12.37864159403511, −11.47539359568889, −10.96791682530584, −10.29915579329998, −10.08156225923280, −9.293785194713515, −8.911913213139615, −8.318545880791698, −7.646250991061884, −6.928786853465612, −6.430432123091344, −5.940951588519136, −5.322372758807355, −4.747384525950520, −3.904337127808482, −3.046340851169198, −2.664694133115335, −2.181195691877715, −0.7322024083033291, 0, 0.7322024083033291, 2.181195691877715, 2.664694133115335, 3.046340851169198, 3.904337127808482, 4.747384525950520, 5.322372758807355, 5.940951588519136, 6.430432123091344, 6.928786853465612, 7.646250991061884, 8.318545880791698, 8.911913213139615, 9.293785194713515, 10.08156225923280, 10.29915579329998, 10.96791682530584, 11.47539359568889, 12.37864159403511, 12.62316671210797, 13.15571969785026, 13.60331584853034, 14.12860792490237, 14.74048752760684, 15.37549537310899

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