Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 7 \cdot 617 $
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 − 3·3-s + 4-s + 5-s − 3·6-s + 7-s + 8-s + 6·9-s + 10-s − 2·11-s − 3·12-s − 7·13-s + 14-s − 3·15-s + 16-s + 4·17-s + 6·18-s − 19-s + 20-s − 3·21-s − 2·22-s + 3·23-s − 3·24-s + 25-s − 7·26-s − 9·27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.316·10-s − 0.603·11-s − 0.866·12-s − 1.94·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.970·17-s + 1.41·18-s − 0.229·19-s + 0.223·20-s − 0.654·21-s − 0.426·22-s + 0.625·23-s − 0.612·24-s + 1/5·25-s − 1.37·26-s − 1.73·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 43190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 43190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43190\)    =    \(2 \cdot 5 \cdot 7 \cdot 617\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{43190} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 43190,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;7,\;617\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7,\;617\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
617 \( 1 - T \)
good3 \( 1 + p T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−14.90763021051958, −14.58923733706531, −13.76434898003083, −13.35452219990939, −12.67906389862674, −12.26550439313926, −11.92143278325472, −11.57292348693169, −10.73664378026602, −10.40319812729528, −9.986808268304155, −9.530438485103528, −8.447241906462556, −7.826896338989122, −7.059092626612508, −6.951574560792965, −6.047247931403473, −5.691183443728905, −5.109392820692658, −4.613228950622470, −4.466623842944050, −3.091180329765601, −2.639360730892436, −1.687626870982717, −0.9468999577608752, 0, 0.9468999577608752, 1.687626870982717, 2.639360730892436, 3.091180329765601, 4.466623842944050, 4.613228950622470, 5.109392820692658, 5.691183443728905, 6.047247931403473, 6.951574560792965, 7.059092626612508, 7.826896338989122, 8.447241906462556, 9.530438485103528, 9.986808268304155, 10.40319812729528, 10.73664378026602, 11.57292348693169, 11.92143278325472, 12.26550439313926, 12.67906389862674, 13.35452219990939, 13.76434898003083, 14.58923733706531, 14.90763021051958

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