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  + 2.46·2-s + 3-s + 4.07·4-s + 0.00469·5-s + 2.46·6-s + 4.87·7-s + 5.11·8-s + 9-s + 0.0115·10-s − 2.31·11-s + 4.07·12-s − 4.34·13-s + 12.0·14-s + 0.00469·15-s + 4.44·16-s + 17-s + 2.46·18-s + 6.19·19-s + 0.0191·20-s + 4.87·21-s − 5.71·22-s + 0.574·23-s + 5.11·24-s − 4.99·25-s − 10.7·26-s + 27-s + 19.8·28-s + ⋯
L(s)  = 1  + 1.74·2-s + 0.577·3-s + 2.03·4-s + 0.00209·5-s + 1.00·6-s + 1.84·7-s + 1.80·8-s + 0.333·9-s + 0.00365·10-s − 0.699·11-s + 1.17·12-s − 1.20·13-s + 3.20·14-s + 0.00121·15-s + 1.11·16-s + 0.242·17-s + 0.580·18-s + 1.42·19-s + 0.00427·20-s + 1.06·21-s − 1.21·22-s + 0.119·23-s + 1.04·24-s − 0.999·25-s − 2.10·26-s + 0.192·27-s + 3.75·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $8.199131787$
$L(\frac12)$  $\approx$  $8.199131787$
$L(\frac{3}{2})$   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 \( 1 - 2.46T + 2T^{2} \)
5 \( 1 - 0.00469T + 5T^{2} \)
7 \( 1 - 4.87T + 7T^{2} \)
11 \( 1 + 2.31T + 11T^{2} \)
13 \( 1 + 4.34T + 13T^{2} \)
19 \( 1 - 6.19T + 19T^{2} \)
23 \( 1 - 0.574T + 23T^{2} \)
29 \( 1 - 1.99T + 29T^{2} \)
31 \( 1 + 1.79T + 31T^{2} \)
37 \( 1 - 4.56T + 37T^{2} \)
41 \( 1 - 4.02T + 41T^{2} \)
43 \( 1 + 4.88T + 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 - 9.71T + 53T^{2} \)
59 \( 1 + 3.03T + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 - 5.89T + 67T^{2} \)
71 \( 1 + 13.1T + 71T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
83 \( 1 + 0.853T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 + 1.53T + 97T^{2} \)
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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

−8.049854620506182944567914041728, −7.52593811308463224658818250879, −7.14597171001984189618651734329, −5.68818975974906192821695911763, −5.39442638666903454907326631733, −4.58731759288648088464162123925, −4.13216624916135207117147069341, −2.94853563780503067205482530772, −2.39031592858489939996164061094, −1.45478959636056631707167215779, 1.45478959636056631707167215779, 2.39031592858489939996164061094, 2.94853563780503067205482530772, 4.13216624916135207117147069341, 4.58731759288648088464162123925, 5.39442638666903454907326631733, 5.68818975974906192821695911763, 7.14597171001984189618651734329, 7.52593811308463224658818250879, 8.049854620506182944567914041728

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