Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 79 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.66·2-s + 3-s + 0.765·4-s + 1.59·5-s + 1.66·6-s − 5.08·7-s − 2.05·8-s + 9-s + 2.64·10-s + 4.95·11-s + 0.765·12-s + 0.606·13-s − 8.46·14-s + 1.59·15-s − 4.94·16-s + 17-s + 1.66·18-s + 3.54·19-s + 1.21·20-s − 5.08·21-s + 8.24·22-s + 4.44·23-s − 2.05·24-s − 2.46·25-s + 1.00·26-s + 27-s − 3.89·28-s + ⋯
L(s)  = 1  + 1.17·2-s + 0.577·3-s + 0.382·4-s + 0.711·5-s + 0.678·6-s − 1.92·7-s − 0.725·8-s + 0.333·9-s + 0.836·10-s + 1.49·11-s + 0.220·12-s + 0.168·13-s − 2.26·14-s + 0.410·15-s − 1.23·16-s + 0.242·17-s + 0.391·18-s + 0.812·19-s + 0.272·20-s − 1.11·21-s + 1.75·22-s + 0.927·23-s − 0.419·24-s − 0.493·25-s + 0.197·26-s + 0.192·27-s − 0.735·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4029\)    =    \(3 \cdot 17 \cdot 79\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4029} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.136158045$
$L(\frac12)$  $\approx$  $4.136158045$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;17,\;79\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;79\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
17 \( 1 - T \)
79 \( 1 - T \)
good2 \( 1 - 1.66T + 2T^{2} \)
5 \( 1 - 1.59T + 5T^{2} \)
7 \( 1 + 5.08T + 7T^{2} \)
11 \( 1 - 4.95T + 11T^{2} \)
13 \( 1 - 0.606T + 13T^{2} \)
19 \( 1 - 3.54T + 19T^{2} \)
23 \( 1 - 4.44T + 23T^{2} \)
29 \( 1 - 8.26T + 29T^{2} \)
31 \( 1 + 5.27T + 31T^{2} \)
37 \( 1 - 7.77T + 37T^{2} \)
41 \( 1 - 1.05T + 41T^{2} \)
43 \( 1 - 7.52T + 43T^{2} \)
47 \( 1 - 3.93T + 47T^{2} \)
53 \( 1 + 5.66T + 53T^{2} \)
59 \( 1 - 1.55T + 59T^{2} \)
61 \( 1 - 7.07T + 61T^{2} \)
67 \( 1 - 7.72T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + 14.8T + 73T^{2} \)
83 \( 1 - 0.297T + 83T^{2} \)
89 \( 1 - 3.46T + 89T^{2} \)
97 \( 1 - 0.671T + 97T^{2} \)
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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

−8.789706633645673974322495673566, −7.38715965731924174402430965013, −6.66967655835075875548583626801, −6.14841525452451202755250463652, −5.62518421209454026796458322278, −4.46460690660535666328121234973, −3.74039712133929531470024450472, −3.16823790680219381228614071171, −2.48526390326967436054771408043, −0.974845110094286214063835361252, 0.974845110094286214063835361252, 2.48526390326967436054771408043, 3.16823790680219381228614071171, 3.74039712133929531470024450472, 4.46460690660535666328121234973, 5.62518421209454026796458322278, 6.14841525452451202755250463652, 6.66967655835075875548583626801, 7.38715965731924174402430965013, 8.789706633645673974322495673566

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