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  − 2.77·2-s − 3-s + 5.71·4-s − 0.322·5-s + 2.77·6-s − 2.70·7-s − 10.3·8-s + 9-s + 0.896·10-s − 4.20·11-s − 5.71·12-s − 3.21·13-s + 7.50·14-s + 0.322·15-s + 17.1·16-s − 17-s − 2.77·18-s + 1.38·19-s − 1.84·20-s + 2.70·21-s + 11.6·22-s − 4.75·23-s + 10.3·24-s − 4.89·25-s + 8.93·26-s − 27-s − 15.4·28-s + ⋯
L(s)  = 1  − 1.96·2-s − 0.577·3-s + 2.85·4-s − 0.144·5-s + 1.13·6-s − 1.02·7-s − 3.64·8-s + 0.333·9-s + 0.283·10-s − 1.26·11-s − 1.64·12-s − 0.892·13-s + 2.00·14-s + 0.0833·15-s + 4.29·16-s − 0.242·17-s − 0.654·18-s + 0.316·19-s − 0.412·20-s + 0.590·21-s + 2.49·22-s − 0.990·23-s + 2.10·24-s − 0.979·25-s + 1.75·26-s − 0.192·27-s − 2.91·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.02272619263$
$L(\frac12)$  $\approx$  $0.02272619263$
$L(\frac{3}{2})$   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 \( 1 + 2.77T + 2T^{2} \)
5 \( 1 + 0.322T + 5T^{2} \)
7 \( 1 + 2.70T + 7T^{2} \)
11 \( 1 + 4.20T + 11T^{2} \)
13 \( 1 + 3.21T + 13T^{2} \)
19 \( 1 - 1.38T + 19T^{2} \)
23 \( 1 + 4.75T + 23T^{2} \)
29 \( 1 - 1.17T + 29T^{2} \)
31 \( 1 - 1.07T + 31T^{2} \)
37 \( 1 + 5.29T + 37T^{2} \)
41 \( 1 + 7.96T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + 3.46T + 53T^{2} \)
59 \( 1 + 9.56T + 59T^{2} \)
61 \( 1 + 12.7T + 61T^{2} \)
67 \( 1 + 6.97T + 67T^{2} \)
71 \( 1 + 5.38T + 71T^{2} \)
73 \( 1 + 8.17T + 73T^{2} \)
83 \( 1 - 6.37T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 - 6.91T + 97T^{2} \)
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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

−8.435713151408237845588535915948, −7.63445022874785997702650086180, −7.38317785996079437191665136869, −6.36110533416784578065617725432, −5.98324009922312854592496562029, −4.91662187102897062293300869876, −3.38514657898121055744959379033, −2.61068976246174767734286385533, −1.68018152191468540767488771880, −0.11302749347556525599494983765, 0.11302749347556525599494983765, 1.68018152191468540767488771880, 2.61068976246174767734286385533, 3.38514657898121055744959379033, 4.91662187102897062293300869876, 5.98324009922312854592496562029, 6.36110533416784578065617725432, 7.38317785996079437191665136869, 7.63445022874785997702650086180, 8.435713151408237845588535915948

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