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  + 2.79·2-s + 3-s + 5.78·4-s + 1.94·5-s + 2.79·6-s − 1.90·7-s + 10.5·8-s + 9-s + 5.42·10-s + 0.728·11-s + 5.78·12-s − 5.07·13-s − 5.32·14-s + 1.94·15-s + 17.9·16-s + 17-s + 2.79·18-s + 2.91·19-s + 11.2·20-s − 1.90·21-s + 2.03·22-s − 1.70·23-s + 10.5·24-s − 1.22·25-s − 14.1·26-s + 27-s − 11.0·28-s + ⋯
L(s)  = 1  + 1.97·2-s + 0.577·3-s + 2.89·4-s + 0.868·5-s + 1.13·6-s − 0.721·7-s + 3.73·8-s + 0.333·9-s + 1.71·10-s + 0.219·11-s + 1.67·12-s − 1.40·13-s − 1.42·14-s + 0.501·15-s + 4.48·16-s + 0.242·17-s + 0.657·18-s + 0.667·19-s + 2.51·20-s − 0.416·21-s + 0.433·22-s − 0.355·23-s + 2.15·24-s − 0.245·25-s − 2.77·26-s + 0.192·27-s − 2.08·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $9.747096921$
$L(\frac12)$  $\approx$  $9.747096921$
$L(\frac{3}{2})$   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 \( 1 - 2.79T + 2T^{2} \)
5 \( 1 - 1.94T + 5T^{2} \)
7 \( 1 + 1.90T + 7T^{2} \)
11 \( 1 - 0.728T + 11T^{2} \)
13 \( 1 + 5.07T + 13T^{2} \)
19 \( 1 - 2.91T + 19T^{2} \)
23 \( 1 + 1.70T + 23T^{2} \)
29 \( 1 - 1.56T + 29T^{2} \)
31 \( 1 - 3.84T + 31T^{2} \)
37 \( 1 - 5.97T + 37T^{2} \)
41 \( 1 + 6.77T + 41T^{2} \)
43 \( 1 + 8.00T + 43T^{2} \)
47 \( 1 + 3.49T + 47T^{2} \)
53 \( 1 + 0.650T + 53T^{2} \)
59 \( 1 - 4.88T + 59T^{2} \)
61 \( 1 + 4.59T + 61T^{2} \)
67 \( 1 + 7.38T + 67T^{2} \)
71 \( 1 + 2.04T + 71T^{2} \)
73 \( 1 - 6.82T + 73T^{2} \)
83 \( 1 - 0.725T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 - 1.34T + 97T^{2} \)
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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

−8.076977607721381956663774029478, −7.42136878539698036302474106955, −6.61680431779893495062882717052, −6.20291392018517120185733860497, −5.26483140794422823914939845013, −4.77389260977519388469662382865, −3.79535923535266586148588491752, −3.03590039567320852183718432402, −2.44736595893419379557230168936, −1.57699285983741823903322209226, 1.57699285983741823903322209226, 2.44736595893419379557230168936, 3.03590039567320852183718432402, 3.79535923535266586148588491752, 4.77389260977519388469662382865, 5.26483140794422823914939845013, 6.20291392018517120185733860497, 6.61680431779893495062882717052, 7.42136878539698036302474106955, 8.076977607721381956663774029478

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