Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 23 \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  − 2-s − 3-s − 4-s − 5-s + 6-s + 3·8-s + 9-s + 10-s − 2·11-s + 12-s + 2·13-s + 15-s − 16-s + 4·17-s − 18-s + 20-s + 2·22-s − 23-s − 3·24-s + 25-s − 2·26-s − 27-s − 30-s + 2·31-s − 5·32-s + 2·33-s − 4·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.554·13-s + 0.258·15-s − 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.223·20-s + 0.426·22-s − 0.208·23-s − 0.612·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.182·30-s + 0.359·31-s − 0.883·32-s + 0.348·33-s − 0.685·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 290145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 290145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(290145\)    =    \(3 \cdot 5 \cdot 23 \cdot 29^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{290145} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 290145,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;23,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
5 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 \)
good2 \( 1 + T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−12.82345045797645, −12.43869732952289, −12.10955408671096, −11.38475056397459, −11.03701637894903, −10.67321691855149, −10.11175607878924, −9.796840251412584, −9.332599860748971, −8.763974072321934, −8.166857914880482, −8.013883959005970, −7.492386699617595, −6.986149394931985, −6.345702554936307, −5.918308530939519, −5.186690022338921, −4.965974600041883, −4.404480621942580, −3.730003617249759, −3.385608995375661, −2.645437079196791, −1.740971082167407, −1.296901293840709, −0.6043207631599170, 0, 0.6043207631599170, 1.296901293840709, 1.740971082167407, 2.645437079196791, 3.385608995375661, 3.730003617249759, 4.404480621942580, 4.965974600041883, 5.186690022338921, 5.918308530939519, 6.345702554936307, 6.986149394931985, 7.492386699617595, 8.013883959005970, 8.166857914880482, 8.763974072321934, 9.332599860748971, 9.796840251412584, 10.11175607878924, 10.67321691855149, 11.03701637894903, 11.38475056397459, 12.10955408671096, 12.43869732952289, 12.82345045797645

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