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  + 0.937·2-s − 3-s − 1.12·4-s + 0.347·5-s − 0.937·6-s + 4.09·7-s − 2.92·8-s + 9-s + 0.325·10-s + 5.59·11-s + 1.12·12-s + 4.86·13-s + 3.84·14-s − 0.347·15-s − 0.502·16-s − 17-s + 0.937·18-s + 2.01·19-s − 0.389·20-s − 4.09·21-s + 5.25·22-s + 7.46·23-s + 2.92·24-s − 4.87·25-s + 4.56·26-s − 27-s − 4.59·28-s + ⋯
L(s)  = 1  + 0.663·2-s − 0.577·3-s − 0.560·4-s + 0.155·5-s − 0.382·6-s + 1.54·7-s − 1.03·8-s + 0.333·9-s + 0.102·10-s + 1.68·11-s + 0.323·12-s + 1.35·13-s + 1.02·14-s − 0.0896·15-s − 0.125·16-s − 0.242·17-s + 0.221·18-s + 0.462·19-s − 0.0869·20-s − 0.893·21-s + 1.11·22-s + 1.55·23-s + 0.597·24-s − 0.975·25-s + 0.895·26-s − 0.192·27-s − 0.867·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.768429446$
$L(\frac12)$  $\approx$  $2.768429446$
$L(\frac{3}{2})$   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 \( 1 - 0.937T + 2T^{2} \)
5 \( 1 - 0.347T + 5T^{2} \)
7 \( 1 - 4.09T + 7T^{2} \)
11 \( 1 - 5.59T + 11T^{2} \)
13 \( 1 - 4.86T + 13T^{2} \)
19 \( 1 - 2.01T + 19T^{2} \)
23 \( 1 - 7.46T + 23T^{2} \)
29 \( 1 + 1.16T + 29T^{2} \)
31 \( 1 + 7.75T + 31T^{2} \)
37 \( 1 - 5.13T + 37T^{2} \)
41 \( 1 - 12.4T + 41T^{2} \)
43 \( 1 + 3.07T + 43T^{2} \)
47 \( 1 + 9.52T + 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 - 4.14T + 59T^{2} \)
61 \( 1 + 4.30T + 61T^{2} \)
67 \( 1 + 1.33T + 67T^{2} \)
71 \( 1 - 5.13T + 71T^{2} \)
73 \( 1 - 8.49T + 73T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 + 14.1T + 97T^{2} \)
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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

−8.501914750988929459876117087965, −7.73266317984856659416941624429, −6.74393491709462252069923431042, −6.00802501350007117679490349903, −5.43375648310874405360950871782, −4.61445307296401334754874954038, −4.09133380396792518340633991382, −3.32244533223071010118007434194, −1.70448183589166599686534802282, −1.01140387182914709440985896253, 1.01140387182914709440985896253, 1.70448183589166599686534802282, 3.32244533223071010118007434194, 4.09133380396792518340633991382, 4.61445307296401334754874954038, 5.43375648310874405360950871782, 6.00802501350007117679490349903, 6.74393491709462252069923431042, 7.73266317984856659416941624429, 8.501914750988929459876117087965

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