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  − 2.65·2-s − 3-s + 5.04·4-s − 3.36·5-s + 2.65·6-s + 2.92·7-s − 8.09·8-s + 9-s + 8.93·10-s + 6.58·11-s − 5.04·12-s − 4.29·13-s − 7.75·14-s + 3.36·15-s + 11.3·16-s − 17-s − 2.65·18-s − 1.04·19-s − 16.9·20-s − 2.92·21-s − 17.4·22-s − 8.38·23-s + 8.09·24-s + 6.32·25-s + 11.4·26-s − 27-s + 14.7·28-s + ⋯
L(s)  = 1  − 1.87·2-s − 0.577·3-s + 2.52·4-s − 1.50·5-s + 1.08·6-s + 1.10·7-s − 2.86·8-s + 0.333·9-s + 2.82·10-s + 1.98·11-s − 1.45·12-s − 1.19·13-s − 2.07·14-s + 0.868·15-s + 2.84·16-s − 0.242·17-s − 0.625·18-s − 0.239·19-s − 3.79·20-s − 0.637·21-s − 3.72·22-s − 1.74·23-s + 1.65·24-s + 1.26·25-s + 2.23·26-s − 0.192·27-s + 2.78·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4470029349$
$L(\frac12)$  $\approx$  $0.4470029349$
$L(\frac{3}{2})$   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 \( 1 + 2.65T + 2T^{2} \)
5 \( 1 + 3.36T + 5T^{2} \)
7 \( 1 - 2.92T + 7T^{2} \)
11 \( 1 - 6.58T + 11T^{2} \)
13 \( 1 + 4.29T + 13T^{2} \)
19 \( 1 + 1.04T + 19T^{2} \)
23 \( 1 + 8.38T + 23T^{2} \)
29 \( 1 + 7.36T + 29T^{2} \)
31 \( 1 - 8.31T + 31T^{2} \)
37 \( 1 - 7.07T + 37T^{2} \)
41 \( 1 - 8.23T + 41T^{2} \)
43 \( 1 - 6.39T + 43T^{2} \)
47 \( 1 - 5.28T + 47T^{2} \)
53 \( 1 - 0.314T + 53T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 + 9.42T + 61T^{2} \)
67 \( 1 - 1.16T + 67T^{2} \)
71 \( 1 + 9.32T + 71T^{2} \)
73 \( 1 + 3.35T + 73T^{2} \)
83 \( 1 + 6.00T + 83T^{2} \)
89 \( 1 + 0.846T + 89T^{2} \)
97 \( 1 - 1.99T + 97T^{2} \)
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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

−8.378096208504794605676195699879, −7.69114781330750268730071243046, −7.43787823956812182116258647520, −6.58729642323399105406337227863, −5.86186928446266747658303215917, −4.38912582393959778051028878203, −4.01995392963389222047531118471, −2.47886856683740211819921146650, −1.48483818119754997633100692765, −0.55600211036384933656828348437, 0.55600211036384933656828348437, 1.48483818119754997633100692765, 2.47886856683740211819921146650, 4.01995392963389222047531118471, 4.38912582393959778051028878203, 5.86186928446266747658303215917, 6.58729642323399105406337227863, 7.43787823956812182116258647520, 7.69114781330750268730071243046, 8.378096208504794605676195699879

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