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  − 1.60·2-s − 3-s + 0.567·4-s − 0.489·5-s + 1.60·6-s − 0.574·7-s + 2.29·8-s + 9-s + 0.784·10-s − 0.698·11-s − 0.567·12-s + 2.26·13-s + 0.919·14-s + 0.489·15-s − 4.81·16-s − 17-s − 1.60·18-s + 1.07·19-s − 0.277·20-s + 0.574·21-s + 1.11·22-s − 7.79·23-s − 2.29·24-s − 4.75·25-s − 3.62·26-s − 27-s − 0.325·28-s + ⋯
L(s)  = 1  − 1.13·2-s − 0.577·3-s + 0.283·4-s − 0.219·5-s + 0.654·6-s − 0.216·7-s + 0.811·8-s + 0.333·9-s + 0.248·10-s − 0.210·11-s − 0.163·12-s + 0.627·13-s + 0.245·14-s + 0.126·15-s − 1.20·16-s − 0.242·17-s − 0.377·18-s + 0.246·19-s − 0.0621·20-s + 0.125·21-s + 0.238·22-s − 1.62·23-s − 0.468·24-s − 0.951·25-s − 0.710·26-s − 0.192·27-s − 0.0615·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5481216112$
$L(\frac12)$  $\approx$  $0.5481216112$
$L(\frac{3}{2})$   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 \( 1 + 1.60T + 2T^{2} \)
5 \( 1 + 0.489T + 5T^{2} \)
7 \( 1 + 0.574T + 7T^{2} \)
11 \( 1 + 0.698T + 11T^{2} \)
13 \( 1 - 2.26T + 13T^{2} \)
19 \( 1 - 1.07T + 19T^{2} \)
23 \( 1 + 7.79T + 23T^{2} \)
29 \( 1 - 5.05T + 29T^{2} \)
31 \( 1 - 9.16T + 31T^{2} \)
37 \( 1 + 0.352T + 37T^{2} \)
41 \( 1 - 9.93T + 41T^{2} \)
43 \( 1 - 7.78T + 43T^{2} \)
47 \( 1 - 2.72T + 47T^{2} \)
53 \( 1 + 3.40T + 53T^{2} \)
59 \( 1 + 6.49T + 59T^{2} \)
61 \( 1 + 3.39T + 61T^{2} \)
67 \( 1 - 3.05T + 67T^{2} \)
71 \( 1 - 3.33T + 71T^{2} \)
73 \( 1 + 6.93T + 73T^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 + 6.31T + 89T^{2} \)
97 \( 1 + 1.51T + 97T^{2} \)
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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

−8.164493226411327908023401328160, −8.089503276612199432050098868170, −7.13850817306353972363787378787, −6.29567041701065957474597383758, −5.71304732587954599426609384244, −4.50585627477543715397285775055, −4.08358261201473934330364966174, −2.72074176279853963425840896387, −1.56703531059202598892769410112, −0.54213775513460817595242206053, 0.54213775513460817595242206053, 1.56703531059202598892769410112, 2.72074176279853963425840896387, 4.08358261201473934330364966174, 4.50585627477543715397285775055, 5.71304732587954599426609384244, 6.29567041701065957474597383758, 7.13850817306353972363787378787, 8.089503276612199432050098868170, 8.164493226411327908023401328160

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