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  − 1.67·2-s − 3-s + 0.810·4-s − 3.83·5-s + 1.67·6-s − 4.72·7-s + 1.99·8-s + 9-s + 6.43·10-s + 6.37·11-s − 0.810·12-s + 6.95·13-s + 7.92·14-s + 3.83·15-s − 4.96·16-s − 17-s − 1.67·18-s + 6.78·19-s − 3.11·20-s + 4.72·21-s − 10.6·22-s + 5.35·23-s − 1.99·24-s + 9.74·25-s − 11.6·26-s − 27-s − 3.83·28-s + ⋯
L(s)  = 1  − 1.18·2-s − 0.577·3-s + 0.405·4-s − 1.71·5-s + 0.684·6-s − 1.78·7-s + 0.704·8-s + 0.333·9-s + 2.03·10-s + 1.92·11-s − 0.234·12-s + 1.92·13-s + 2.11·14-s + 0.991·15-s − 1.24·16-s − 0.242·17-s − 0.395·18-s + 1.55·19-s − 0.695·20-s + 1.03·21-s − 2.27·22-s + 1.11·23-s − 0.407·24-s + 1.94·25-s − 2.28·26-s − 0.192·27-s − 0.724·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5424873365$
$L(\frac12)$  $\approx$  $0.5424873365$
$L(\frac{3}{2})$   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 \( 1 + 1.67T + 2T^{2} \)
5 \( 1 + 3.83T + 5T^{2} \)
7 \( 1 + 4.72T + 7T^{2} \)
11 \( 1 - 6.37T + 11T^{2} \)
13 \( 1 - 6.95T + 13T^{2} \)
19 \( 1 - 6.78T + 19T^{2} \)
23 \( 1 - 5.35T + 23T^{2} \)
29 \( 1 + 3.88T + 29T^{2} \)
31 \( 1 + 4.37T + 31T^{2} \)
37 \( 1 + 1.41T + 37T^{2} \)
41 \( 1 - 6.61T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 - 3.03T + 47T^{2} \)
53 \( 1 + 3.69T + 53T^{2} \)
59 \( 1 + 0.276T + 59T^{2} \)
61 \( 1 - 4.92T + 61T^{2} \)
67 \( 1 + 4.08T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 - 1.10T + 73T^{2} \)
83 \( 1 + 3.30T + 83T^{2} \)
89 \( 1 - 9.92T + 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−8.791493033730995340283055467893, −7.57844155987714965275401950133, −7.17037637111665214144050306240, −6.52261902214139315657403288052, −5.78444301915067707139070564270, −4.32930116863224973723948913070, −3.75919765903948967609821596808, −3.28509198689480904048877010075, −1.17753141718733857387006867298, −0.65315778291841160113057828433, 0.65315778291841160113057828433, 1.17753141718733857387006867298, 3.28509198689480904048877010075, 3.75919765903948967609821596808, 4.32930116863224973723948913070, 5.78444301915067707139070564270, 6.52261902214139315657403288052, 7.17037637111665214144050306240, 7.57844155987714965275401950133, 8.791493033730995340283055467893

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