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  − 2.41·2-s + 3-s + 3.84·4-s + 1.43·5-s − 2.41·6-s + 3.74·7-s − 4.46·8-s + 9-s − 3.47·10-s + 4.27·11-s + 3.84·12-s + 2.58·13-s − 9.04·14-s + 1.43·15-s + 3.11·16-s + 17-s − 2.41·18-s − 7.71·19-s + 5.52·20-s + 3.74·21-s − 10.3·22-s + 3.97·23-s − 4.46·24-s − 2.93·25-s − 6.26·26-s + 27-s + 14.3·28-s + ⋯
L(s)  = 1  − 1.70·2-s + 0.577·3-s + 1.92·4-s + 0.642·5-s − 0.987·6-s + 1.41·7-s − 1.58·8-s + 0.333·9-s − 1.09·10-s + 1.28·11-s + 1.11·12-s + 0.718·13-s − 2.41·14-s + 0.371·15-s + 0.777·16-s + 0.242·17-s − 0.569·18-s − 1.76·19-s + 1.23·20-s + 0.816·21-s − 2.20·22-s + 0.828·23-s − 0.912·24-s − 0.586·25-s − 1.22·26-s + 0.192·27-s + 2.72·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.658826964$
$L(\frac12)$  $\approx$  $1.658826964$
$L(\frac{3}{2})$   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 \( 1 + 2.41T + 2T^{2} \)
5 \( 1 - 1.43T + 5T^{2} \)
7 \( 1 - 3.74T + 7T^{2} \)
11 \( 1 - 4.27T + 11T^{2} \)
13 \( 1 - 2.58T + 13T^{2} \)
19 \( 1 + 7.71T + 19T^{2} \)
23 \( 1 - 3.97T + 23T^{2} \)
29 \( 1 + 2.55T + 29T^{2} \)
31 \( 1 + 1.53T + 31T^{2} \)
37 \( 1 + 2.68T + 37T^{2} \)
41 \( 1 - 8.65T + 41T^{2} \)
43 \( 1 + 2.21T + 43T^{2} \)
47 \( 1 - 2.63T + 47T^{2} \)
53 \( 1 + 0.811T + 53T^{2} \)
59 \( 1 - 8.80T + 59T^{2} \)
61 \( 1 - 1.53T + 61T^{2} \)
67 \( 1 - 2.18T + 67T^{2} \)
71 \( 1 + 1.36T + 71T^{2} \)
73 \( 1 - 3.33T + 73T^{2} \)
83 \( 1 - 0.585T + 83T^{2} \)
89 \( 1 - 10.0T + 89T^{2} \)
97 \( 1 - 1.60T + 97T^{2} \)
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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

−8.611878732064763427601921228367, −8.016740513118055753863879294977, −7.25326753743597278011943407266, −6.54180594975180239016267479660, −5.79998805256832890888376298248, −4.57796813242387442708724149966, −3.72570696904172611441269523559, −2.27913519340406835992657473196, −1.77598952128462511039103848836, −1.01381305004215652564410974069, 1.01381305004215652564410974069, 1.77598952128462511039103848836, 2.27913519340406835992657473196, 3.72570696904172611441269523559, 4.57796813242387442708724149966, 5.79998805256832890888376298248, 6.54180594975180239016267479660, 7.25326753743597278011943407266, 8.016740513118055753863879294977, 8.611878732064763427601921228367

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