Properties

Degree 2
Conductor 18097
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 4-s − 4·5-s + 6-s − 7-s + 3·8-s − 2·9-s + 4·10-s − 6·11-s + 12-s − 5·13-s + 14-s + 4·15-s − 16-s − 7·17-s + 2·18-s − 4·19-s + 4·20-s + 21-s + 6·22-s − 4·23-s − 3·24-s + 11·25-s + 5·26-s + 5·27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s − 2/3·9-s + 1.26·10-s − 1.80·11-s + 0.288·12-s − 1.38·13-s + 0.267·14-s + 1.03·15-s − 1/4·16-s − 1.69·17-s + 0.471·18-s − 0.917·19-s + 0.894·20-s + 0.218·21-s + 1.27·22-s − 0.834·23-s − 0.612·24-s + 11/5·25-s + 0.980·26-s + 0.962·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 18097 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 18097 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(18097\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{18097} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 18097,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 18097$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 18097$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad18097 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−16.66355078553868, −16.12723187140413, −15.66465705717195, −15.08798951136561, −14.69033501166069, −13.80715341856277, −13.12891148169160, −12.62186029198830, −12.31386071185314, −11.31674063975217, −11.12474175457839, −10.59241454938144, −9.972604400204676, −9.235211865855686, −8.578833682213959, −8.093201974911126, −7.703540173776869, −7.120167421868357, −6.410646553189068, −5.409430294898282, −4.732797424415094, −4.536545453925919, −3.577472295292634, −2.801843415547256, −1.946680073210762, 0, 0, 0, 1.946680073210762, 2.801843415547256, 3.577472295292634, 4.536545453925919, 4.732797424415094, 5.409430294898282, 6.410646553189068, 7.120167421868357, 7.703540173776869, 8.093201974911126, 8.578833682213959, 9.235211865855686, 9.972604400204676, 10.59241454938144, 11.12474175457839, 11.31674063975217, 12.31386071185314, 12.62186029198830, 13.12891148169160, 13.80715341856277, 14.69033501166069, 15.08798951136561, 15.66465705717195, 16.12723187140413, 16.66355078553868

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