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 + 0.585·11-s + 1.41·12-s − 1.82·13-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 + 0.585·22-s − 5.07·23-s + 1.41·24-s + 25-s − 1.82·26-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 + 0.176·11-s + 0.408·12-s − 0.507·13-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 + 0.124·22-s − 1.05·23-s + 0.288·24-s + 0.200·25-s − 0.358·26-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  =  $(2,\ 430,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.77723$
$L(\frac12)$  $\approx$  $2.77723$
$L(\frac{3}{2})$   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 \( 1 - 0.585T + 11T^{2} \)
13 \( 1 + 1.82T + 13T^{2} \)
17 \( 1 - 1.41T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + 5.07T + 23T^{2} \)
29 \( 1 - 1.24T + 29T^{2} \)
31 \( 1 + 0.414T + 31T^{2} \)
37 \( 1 - 2.24T + 37T^{2} \)
41 \( 1 + 1.82T + 41T^{2} \)
47 \( 1 - 7.07T + 47T^{2} \)
53 \( 1 + 5.65T + 53T^{2} \)
59 \( 1 + 1.17T + 59T^{2} \)
61 \( 1 - 0.0710T + 61T^{2} \)
67 \( 1 + 1.24T + 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 + 9.24T + 73T^{2} \)
79 \( 1 + 8.41T + 79T^{2} \)
83 \( 1 - 3.65T + 83T^{2} \)
89 \( 1 + 9.07T + 89T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−11.30624787575493870781868277133, −10.24098341812761378480223771638, −9.351094299492888227524144359285, −8.335380592868157649115040679983, −7.54034187101934031417881441002, −6.31136087914550595721001935444, −5.37469575261544498847155120468, −4.20580724923551974763393766994, −3.01239367251456204280302943165, −1.95870692717463030361380630541, 1.95870692717463030361380630541, 3.01239367251456204280302943165, 4.20580724923551974763393766994, 5.37469575261544498847155120468, 6.31136087914550595721001935444, 7.54034187101934031417881441002, 8.335380592868157649115040679983, 9.351094299492888227524144359285, 10.24098341812761378480223771638, 11.30624787575493870781868277133

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