Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11 \cdot 19^{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 − 4-s + 2·5-s + 7-s + 3·8-s − 2·10-s + 11-s − 6·13-s − 14-s − 16-s − 2·17-s − 2·20-s − 22-s − 25-s + 6·26-s − 28-s − 2·29-s − 8·31-s − 5·32-s + 2·34-s + 2·35-s − 6·37-s + 6·40-s + 10·41-s − 4·43-s − 44-s + 8·47-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s + 1.06·8-s − 0.632·10-s + 0.301·11-s − 1.66·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.447·20-s − 0.213·22-s − 1/5·25-s + 1.17·26-s − 0.188·28-s − 0.371·29-s − 1.43·31-s − 0.883·32-s + 0.342·34-s + 0.338·35-s − 0.986·37-s + 0.948·40-s + 1.56·41-s − 0.609·43-s − 0.150·44-s + 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(250173\)    =    \(3^{2} \cdot 7 \cdot 11 \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{250173} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 250173,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
19 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.04820991121984, −12.63037060661755, −12.27652087628740, −11.60992026782373, −11.10485746781808, −10.64854131827015, −10.09899379937600, −9.855545298439218, −9.315347804574863, −8.972908115431009, −8.661926371103554, −7.824444260179570, −7.545882672839475, −7.100249711260111, −6.585420250506793, −5.758181632010456, −5.421100092825427, −5.018801362904949, −4.326437739591212, −4.025817970198569, −3.214807752632724, −2.353986043949581, −2.052604558848981, −1.504294291789919, −0.6759407519195875, 0, 0.6759407519195875, 1.504294291789919, 2.052604558848981, 2.353986043949581, 3.214807752632724, 4.025817970198569, 4.326437739591212, 5.018801362904949, 5.421100092825427, 5.758181632010456, 6.585420250506793, 7.100249711260111, 7.545882672839475, 7.824444260179570, 8.661926371103554, 8.972908115431009, 9.315347804574863, 9.855545298439218, 10.09899379937600, 10.64854131827015, 11.10485746781808, 11.60992026782373, 12.27652087628740, 12.63037060661755, 13.04820991121984

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