Properties

Degree 2
Conductor 22481
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 − 4-s − 3·5-s − 4·7-s + 3·8-s − 3·9-s + 3·10-s − 5·11-s − 5·13-s + 4·14-s − 16-s − 6·17-s + 3·18-s − 2·19-s + 3·20-s + 5·22-s − 5·23-s + 4·25-s + 5·26-s + 4·28-s − 6·29-s − 31-s − 5·32-s + 6·34-s + 12·35-s + 3·36-s − 10·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 1.34·5-s − 1.51·7-s + 1.06·8-s − 9-s + 0.948·10-s − 1.50·11-s − 1.38·13-s + 1.06·14-s − 1/4·16-s − 1.45·17-s + 0.707·18-s − 0.458·19-s + 0.670·20-s + 1.06·22-s − 1.04·23-s + 4/5·25-s + 0.980·26-s + 0.755·28-s − 1.11·29-s − 0.179·31-s − 0.883·32-s + 1.02·34-s + 2.02·35-s + 1/2·36-s − 1.64·37-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 22481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 22481 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(22481\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{22481} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 22481,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 22481$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 22481$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad22481 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−16.31412744680317, −15.84506486430773, −15.40089152022480, −14.88564954708737, −14.13729513875034, −13.46695896171871, −13.04992604817202, −12.55223384962710, −12.00169070414373, −11.31201966340766, −10.75600056824182, −10.16598770995452, −9.734495038688310, −9.088355321211834, −8.431163087103359, −8.139967942913953, −7.456894954199839, −6.956473617631122, −6.289762735567401, −5.142862181184030, −5.052613258470793, −3.939008856989872, −3.565908922486521, −2.684976593361542, −2.061822175796283, 0, 0, 0, 2.061822175796283, 2.684976593361542, 3.565908922486521, 3.939008856989872, 5.052613258470793, 5.142862181184030, 6.289762735567401, 6.956473617631122, 7.456894954199839, 8.139967942913953, 8.431163087103359, 9.088355321211834, 9.734495038688310, 10.16598770995452, 10.75600056824182, 11.31201966340766, 12.00169070414373, 12.55223384962710, 13.04992604817202, 13.46695896171871, 14.13729513875034, 14.88564954708737, 15.40089152022480, 15.84506486430773, 16.31412744680317

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