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  − 2.50·2-s − 3-s + 4.29·4-s − 2.69·5-s + 2.50·6-s + 3.74·7-s − 5.75·8-s + 9-s + 6.75·10-s − 1.71·11-s − 4.29·12-s − 0.139·13-s − 9.39·14-s + 2.69·15-s + 5.84·16-s − 17-s − 2.50·18-s + 5.54·19-s − 11.5·20-s − 3.74·21-s + 4.31·22-s + 5.39·23-s + 5.75·24-s + 2.25·25-s + 0.348·26-s − 27-s + 16.0·28-s + ⋯
L(s)  = 1  − 1.77·2-s − 0.577·3-s + 2.14·4-s − 1.20·5-s + 1.02·6-s + 1.41·7-s − 2.03·8-s + 0.333·9-s + 2.13·10-s − 0.518·11-s − 1.23·12-s − 0.0385·13-s − 2.51·14-s + 0.695·15-s + 1.46·16-s − 0.242·17-s − 0.591·18-s + 1.27·19-s − 2.58·20-s − 0.817·21-s + 0.919·22-s + 1.12·23-s + 1.17·24-s + 0.451·25-s + 0.0683·26-s − 0.192·27-s + 3.03·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5462459790$
$L(\frac12)$  $\approx$  $0.5462459790$
$L(\frac{3}{2})$   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 \( 1 + 2.50T + 2T^{2} \)
5 \( 1 + 2.69T + 5T^{2} \)
7 \( 1 - 3.74T + 7T^{2} \)
11 \( 1 + 1.71T + 11T^{2} \)
13 \( 1 + 0.139T + 13T^{2} \)
19 \( 1 - 5.54T + 19T^{2} \)
23 \( 1 - 5.39T + 23T^{2} \)
29 \( 1 - 7.20T + 29T^{2} \)
31 \( 1 + 6.35T + 31T^{2} \)
37 \( 1 + 9.56T + 37T^{2} \)
41 \( 1 - 0.327T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + 8.65T + 47T^{2} \)
53 \( 1 - 2.07T + 53T^{2} \)
59 \( 1 - 2.59T + 59T^{2} \)
61 \( 1 - 15.1T + 61T^{2} \)
67 \( 1 - 4.21T + 67T^{2} \)
71 \( 1 - 8.71T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
83 \( 1 + 16.7T + 83T^{2} \)
89 \( 1 + 0.176T + 89T^{2} \)
97 \( 1 + 9.72T + 97T^{2} \)
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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

−8.327585979063292052776473446381, −7.88121011358021562544640943651, −7.24074132055481572508947995511, −6.80166524226348717477883336797, −5.42598374205800545433208511289, −4.88430491762643286978550541544, −3.75243189918396183267124321673, −2.57299047043661570138872444982, −1.44701979622222594704145480093, −0.61829842328492096045704937369, 0.61829842328492096045704937369, 1.44701979622222594704145480093, 2.57299047043661570138872444982, 3.75243189918396183267124321673, 4.88430491762643286978550541544, 5.42598374205800545433208511289, 6.80166524226348717477883336797, 7.24074132055481572508947995511, 7.88121011358021562544640943651, 8.327585979063292052776473446381

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