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  − 2.03·2-s − 3-s + 2.12·4-s + 1.66·5-s + 2.03·6-s − 1.52·7-s − 0.260·8-s + 9-s − 3.39·10-s − 3.97·11-s − 2.12·12-s − 5.52·13-s + 3.09·14-s − 1.66·15-s − 3.72·16-s − 17-s − 2.03·18-s − 3.49·19-s + 3.55·20-s + 1.52·21-s + 8.07·22-s − 6.50·23-s + 0.260·24-s − 2.21·25-s + 11.2·26-s − 27-s − 3.24·28-s + ⋯
L(s)  = 1  − 1.43·2-s − 0.577·3-s + 1.06·4-s + 0.746·5-s + 0.829·6-s − 0.576·7-s − 0.0921·8-s + 0.333·9-s − 1.07·10-s − 1.19·11-s − 0.614·12-s − 1.53·13-s + 0.827·14-s − 0.431·15-s − 0.931·16-s − 0.242·17-s − 0.478·18-s − 0.802·19-s + 0.794·20-s + 0.332·21-s + 1.72·22-s − 1.35·23-s + 0.0531·24-s − 0.442·25-s + 2.20·26-s − 0.192·27-s − 0.613·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.1328842834$
$L(\frac12)$  $\approx$  $0.1328842834$
$L(\frac{3}{2})$   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 \( 1 + 2.03T + 2T^{2} \)
5 \( 1 - 1.66T + 5T^{2} \)
7 \( 1 + 1.52T + 7T^{2} \)
11 \( 1 + 3.97T + 11T^{2} \)
13 \( 1 + 5.52T + 13T^{2} \)
19 \( 1 + 3.49T + 19T^{2} \)
23 \( 1 + 6.50T + 23T^{2} \)
29 \( 1 + 9.08T + 29T^{2} \)
31 \( 1 + 3.43T + 31T^{2} \)
37 \( 1 + 0.355T + 37T^{2} \)
41 \( 1 + 8.42T + 41T^{2} \)
43 \( 1 + 6.82T + 43T^{2} \)
47 \( 1 - 13.4T + 47T^{2} \)
53 \( 1 + 9.28T + 53T^{2} \)
59 \( 1 - 3.98T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 16.6T + 73T^{2} \)
83 \( 1 + 11.9T + 83T^{2} \)
89 \( 1 + 6.77T + 89T^{2} \)
97 \( 1 + 4.06T + 97T^{2} \)
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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

−8.433285614591560330833594328169, −7.79472509156870428946878670958, −7.12647964950674870442615207679, −6.46344357534835942241938735902, −5.54667401812346088167047852181, −4.96692409906376985753697001214, −3.79567620708206109592874083512, −2.25599040864860696933186118266, −2.01591522951575852108814220595, −0.24769022010946652530868098900, 0.24769022010946652530868098900, 2.01591522951575852108814220595, 2.25599040864860696933186118266, 3.79567620708206109592874083512, 4.96692409906376985753697001214, 5.54667401812346088167047852181, 6.46344357534835942241938735902, 7.12647964950674870442615207679, 7.79472509156870428946878670958, 8.433285614591560330833594328169

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