Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 43 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.41·3-s + 4-s + 5-s − 1.41·6-s + 7-s + 8-s − 0.999·9-s + 10-s + 3.41·11-s − 1.41·12-s + 3.82·13-s + 14-s − 1.41·15-s + 16-s − 1.41·17-s − 0.999·18-s − 19-s + 20-s − 1.41·21-s + 3.41·22-s + 9.07·23-s − 1.41·24-s + 25-s + 3.82·26-s + 5.65·27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.816·3-s + 0.5·4-s + 0.447·5-s − 0.577·6-s + 0.377·7-s + 0.353·8-s − 0.333·9-s + 0.316·10-s + 1.02·11-s − 0.408·12-s + 1.06·13-s + 0.267·14-s − 0.365·15-s + 0.250·16-s − 0.342·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s − 0.308·21-s + 0.727·22-s + 1.89·23-s − 0.288·24-s + 0.200·25-s + 0.750·26-s + 1.08·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(430\)    =    \(2 \cdot 5 \cdot 43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{430} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 430,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.88247$
$L(\frac12)$  $\approx$  $1.88247$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
43 \( 1 - T \)
good3 \( 1 + 1.41T + 3T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 - 3.41T + 11T^{2} \)
13 \( 1 - 3.82T + 13T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 - 9.07T + 23T^{2} \)
29 \( 1 + 7.24T + 29T^{2} \)
31 \( 1 - 2.41T + 31T^{2} \)
37 \( 1 + 6.24T + 37T^{2} \)
41 \( 1 - 3.82T + 41T^{2} \)
47 \( 1 + 7.07T + 47T^{2} \)
53 \( 1 - 5.65T + 53T^{2} \)
59 \( 1 + 6.82T + 59T^{2} \)
61 \( 1 + 14.0T + 61T^{2} \)
67 \( 1 - 7.24T + 67T^{2} \)
71 \( 1 + 7.89T + 71T^{2} \)
73 \( 1 + 0.757T + 73T^{2} \)
79 \( 1 + 5.58T + 79T^{2} \)
83 \( 1 + 7.65T + 83T^{2} \)
89 \( 1 - 5.07T + 89T^{2} \)
97 \( 1 + 17.5T + 97T^{2} \)
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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

−11.13334187713627914064632994170, −10.81183372735846856091763662626, −9.334299586104098062957013815812, −8.510999939426121827868389744923, −7.02615058530844605478785323088, −6.25760926357992075675274873255, −5.47706094261517588503871830415, −4.48870149271777877643396807300, −3.19624398199023331230385161918, −1.45877734356199781399905351842, 1.45877734356199781399905351842, 3.19624398199023331230385161918, 4.48870149271777877643396807300, 5.47706094261517588503871830415, 6.25760926357992075675274873255, 7.02615058530844605478785323088, 8.510999939426121827868389744923, 9.334299586104098062957013815812, 10.81183372735846856091763662626, 11.13334187713627914064632994170

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