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  + 0.617·2-s + 3-s − 1.61·4-s + 3.16·5-s + 0.617·6-s − 0.196·7-s − 2.23·8-s + 9-s + 1.95·10-s − 1.60·11-s − 1.61·12-s − 1.04·13-s − 0.121·14-s + 3.16·15-s + 1.85·16-s + 17-s + 0.617·18-s − 1.16·19-s − 5.12·20-s − 0.196·21-s − 0.990·22-s − 0.115·23-s − 2.23·24-s + 5.02·25-s − 0.645·26-s + 27-s + 0.317·28-s + ⋯
L(s)  = 1  + 0.436·2-s + 0.577·3-s − 0.809·4-s + 1.41·5-s + 0.252·6-s − 0.0741·7-s − 0.790·8-s + 0.333·9-s + 0.618·10-s − 0.483·11-s − 0.467·12-s − 0.289·13-s − 0.0323·14-s + 0.817·15-s + 0.463·16-s + 0.242·17-s + 0.145·18-s − 0.267·19-s − 1.14·20-s − 0.0428·21-s − 0.211·22-s − 0.0239·23-s − 0.456·24-s + 1.00·25-s − 0.126·26-s + 0.192·27-s + 0.0599·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.024253973$
$L(\frac12)$  $\approx$  $3.024253973$
$L(\frac{3}{2})$   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 \( 1 - 0.617T + 2T^{2} \)
5 \( 1 - 3.16T + 5T^{2} \)
7 \( 1 + 0.196T + 7T^{2} \)
11 \( 1 + 1.60T + 11T^{2} \)
13 \( 1 + 1.04T + 13T^{2} \)
19 \( 1 + 1.16T + 19T^{2} \)
23 \( 1 + 0.115T + 23T^{2} \)
29 \( 1 - 6.34T + 29T^{2} \)
31 \( 1 + 1.71T + 31T^{2} \)
37 \( 1 - 6.10T + 37T^{2} \)
41 \( 1 - 3.43T + 41T^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 + 2.31T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 - 2.43T + 59T^{2} \)
61 \( 1 + 1.11T + 61T^{2} \)
67 \( 1 - 9.11T + 67T^{2} \)
71 \( 1 - 7.31T + 71T^{2} \)
73 \( 1 - 7.19T + 73T^{2} \)
83 \( 1 - 1.78T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 - 2.57T + 97T^{2} \)
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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

−8.494046423738194985056547476625, −7.895761539231300125610160295777, −6.84504969714356246472538161643, −6.02760841885420834830728992954, −5.45621051122344183323338622161, −4.71413825063157451278019556881, −3.90572590780503673658643608901, −2.83962832001285806424296773206, −2.25776415176919560723921108021, −0.932445305102377252574445124722, 0.932445305102377252574445124722, 2.25776415176919560723921108021, 2.83962832001285806424296773206, 3.90572590780503673658643608901, 4.71413825063157451278019556881, 5.45621051122344183323338622161, 6.02760841885420834830728992954, 6.84504969714356246472538161643, 7.895761539231300125610160295777, 8.494046423738194985056547476625

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