Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 17^{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 − 4·5-s − 7-s − 3·8-s − 4·10-s − 2·13-s − 14-s − 16-s − 7·19-s + 4·20-s − 4·23-s + 11·25-s − 2·26-s + 28-s − 3·29-s + 7·31-s + 5·32-s + 4·35-s + 10·37-s − 7·38-s + 12·40-s − 2·43-s − 4·46-s − 3·47-s + 49-s + 11·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.78·5-s − 0.377·7-s − 1.06·8-s − 1.26·10-s − 0.554·13-s − 0.267·14-s − 1/4·16-s − 1.60·19-s + 0.894·20-s − 0.834·23-s + 11/5·25-s − 0.392·26-s + 0.188·28-s − 0.557·29-s + 1.25·31-s + 0.883·32-s + 0.676·35-s + 1.64·37-s − 1.13·38-s + 1.89·40-s − 0.304·43-s − 0.589·46-s − 0.437·47-s + 1/7·49-s + 1.55·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(18207\)    =    \(3^{2} \cdot 7 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{18207} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 18207,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
17 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
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

−15.99196754286716, −15.20938564616589, −15.03745595240692, −14.59432416218976, −13.89300335147667, −13.13034110395076, −12.77062063938769, −12.31772439912574, −11.57173711982817, −11.52285704418950, −10.47521116828508, −10.01065379032745, −9.172156507841854, −8.635385272001833, −8.007768812575880, −7.700450132942973, −6.667198691084774, −6.357953588392160, −5.402112414077336, −4.686147509217060, −4.125308752691945, −3.876911113753477, −3.021999539820589, −2.333146905558623, −0.7238673612340793, 0, 0.7238673612340793, 2.333146905558623, 3.021999539820589, 3.876911113753477, 4.125308752691945, 4.686147509217060, 5.402112414077336, 6.357953588392160, 6.667198691084774, 7.700450132942973, 8.007768812575880, 8.635385272001833, 9.172156507841854, 10.01065379032745, 10.47521116828508, 11.52285704418950, 11.57173711982817, 12.31772439912574, 12.77062063938769, 13.13034110395076, 13.89300335147667, 14.59432416218976, 15.03745595240692, 15.20938564616589, 15.99196754286716

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