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  − 0.857·2-s − 3-s − 1.26·4-s − 4.41·5-s + 0.857·6-s + 0.00991·7-s + 2.79·8-s + 9-s + 3.78·10-s − 3.26·11-s + 1.26·12-s + 2.70·13-s − 0.00850·14-s + 4.41·15-s + 0.126·16-s − 17-s − 0.857·18-s − 4.27·19-s + 5.58·20-s − 0.00991·21-s + 2.80·22-s − 8.57·23-s − 2.79·24-s + 14.5·25-s − 2.31·26-s − 27-s − 0.0125·28-s + ⋯
L(s)  = 1  − 0.606·2-s − 0.577·3-s − 0.632·4-s − 1.97·5-s + 0.350·6-s + 0.00374·7-s + 0.989·8-s + 0.333·9-s + 1.19·10-s − 0.984·11-s + 0.364·12-s + 0.749·13-s − 0.00227·14-s + 1.14·15-s + 0.0316·16-s − 0.242·17-s − 0.202·18-s − 0.980·19-s + 1.24·20-s − 0.00216·21-s + 0.597·22-s − 1.78·23-s − 0.571·24-s + 2.90·25-s − 0.454·26-s − 0.192·27-s − 0.00236·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.005658811546$
$L(\frac12)$  $\approx$  $0.005658811546$
$L(\frac{3}{2})$   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 \( 1 + 0.857T + 2T^{2} \)
5 \( 1 + 4.41T + 5T^{2} \)
7 \( 1 - 0.00991T + 7T^{2} \)
11 \( 1 + 3.26T + 11T^{2} \)
13 \( 1 - 2.70T + 13T^{2} \)
19 \( 1 + 4.27T + 19T^{2} \)
23 \( 1 + 8.57T + 23T^{2} \)
29 \( 1 - 2.58T + 29T^{2} \)
31 \( 1 + 3.29T + 31T^{2} \)
37 \( 1 + 5.23T + 37T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 - 1.43T + 43T^{2} \)
47 \( 1 + 1.69T + 47T^{2} \)
53 \( 1 + 4.25T + 53T^{2} \)
59 \( 1 - 0.575T + 59T^{2} \)
61 \( 1 + 3.22T + 61T^{2} \)
67 \( 1 + 7.36T + 67T^{2} \)
71 \( 1 + 15.6T + 71T^{2} \)
73 \( 1 + 6.91T + 73T^{2} \)
83 \( 1 - 0.651T + 83T^{2} \)
89 \( 1 + 7.54T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−8.367876641615097576738676298289, −7.903322144005323976785851536902, −7.24791846590248825384745494316, −6.38396740655429920235470735222, −5.30726334016021227745101574566, −4.49778737315380567411703724714, −4.04822430573724579838247397414, −3.19842422126770713752897105664, −1.57481477815035479204691276279, −0.04968889666658495381935536458, 0.04968889666658495381935536458, 1.57481477815035479204691276279, 3.19842422126770713752897105664, 4.04822430573724579838247397414, 4.49778737315380567411703724714, 5.30726334016021227745101574566, 6.38396740655429920235470735222, 7.24791846590248825384745494316, 7.903322144005323976785851536902, 8.367876641615097576738676298289

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