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  + 2.54·2-s + 3-s + 4.48·4-s + 0.532·5-s + 2.54·6-s + 0.266·7-s + 6.32·8-s + 9-s + 1.35·10-s + 3.13·11-s + 4.48·12-s + 4.93·13-s + 0.677·14-s + 0.532·15-s + 7.13·16-s + 17-s + 2.54·18-s − 4.75·19-s + 2.38·20-s + 0.266·21-s + 7.98·22-s − 4.73·23-s + 6.32·24-s − 4.71·25-s + 12.5·26-s + 27-s + 1.19·28-s + ⋯
L(s)  = 1  + 1.80·2-s + 0.577·3-s + 2.24·4-s + 0.238·5-s + 1.03·6-s + 0.100·7-s + 2.23·8-s + 0.333·9-s + 0.428·10-s + 0.946·11-s + 1.29·12-s + 1.36·13-s + 0.181·14-s + 0.137·15-s + 1.78·16-s + 0.242·17-s + 0.600·18-s − 1.09·19-s + 0.533·20-s + 0.0580·21-s + 1.70·22-s − 0.988·23-s + 1.29·24-s − 0.943·25-s + 2.46·26-s + 0.192·27-s + 0.225·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $8.510938743$
$L(\frac12)$  $\approx$  $8.510938743$
$L(\frac{3}{2})$   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 \( 1 - 2.54T + 2T^{2} \)
5 \( 1 - 0.532T + 5T^{2} \)
7 \( 1 - 0.266T + 7T^{2} \)
11 \( 1 - 3.13T + 11T^{2} \)
13 \( 1 - 4.93T + 13T^{2} \)
19 \( 1 + 4.75T + 19T^{2} \)
23 \( 1 + 4.73T + 23T^{2} \)
29 \( 1 - 1.01T + 29T^{2} \)
31 \( 1 + 5.44T + 31T^{2} \)
37 \( 1 - 5.22T + 37T^{2} \)
41 \( 1 - 2.21T + 41T^{2} \)
43 \( 1 - 7.57T + 43T^{2} \)
47 \( 1 + 3.64T + 47T^{2} \)
53 \( 1 + 8.29T + 53T^{2} \)
59 \( 1 - 7.56T + 59T^{2} \)
61 \( 1 + 2.77T + 61T^{2} \)
67 \( 1 - 5.37T + 67T^{2} \)
71 \( 1 + 6.65T + 71T^{2} \)
73 \( 1 - 1.86T + 73T^{2} \)
83 \( 1 - 7.57T + 83T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 - 7.15T + 97T^{2} \)
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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

−8.264708255278791755581675830086, −7.56245496559848537523652589629, −6.50976633957829567598739962188, −6.20504769753552515166896357146, −5.49178660617728990807803717101, −4.33666317669393125068677428606, −3.97357859391364176123722271116, −3.28856726841288010255820093675, −2.22250012929468804553102722357, −1.51046941928847910118595587737, 1.51046941928847910118595587737, 2.22250012929468804553102722357, 3.28856726841288010255820093675, 3.97357859391364176123722271116, 4.33666317669393125068677428606, 5.49178660617728990807803717101, 6.20504769753552515166896357146, 6.50976633957829567598739962188, 7.56245496559848537523652589629, 8.264708255278791755581675830086

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