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  + 1.90·2-s − 3-s + 1.62·4-s − 1.74·5-s − 1.90·6-s − 2.76·7-s − 0.715·8-s + 9-s − 3.33·10-s − 4.86·11-s − 1.62·12-s + 4.21·13-s − 5.25·14-s + 1.74·15-s − 4.61·16-s − 17-s + 1.90·18-s − 2.06·19-s − 2.84·20-s + 2.76·21-s − 9.25·22-s + 6.78·23-s + 0.715·24-s − 1.93·25-s + 8.02·26-s − 27-s − 4.48·28-s + ⋯
L(s)  = 1  + 1.34·2-s − 0.577·3-s + 0.811·4-s − 0.782·5-s − 0.777·6-s − 1.04·7-s − 0.253·8-s + 0.333·9-s − 1.05·10-s − 1.46·11-s − 0.468·12-s + 1.16·13-s − 1.40·14-s + 0.451·15-s − 1.15·16-s − 0.242·17-s + 0.448·18-s − 0.473·19-s − 0.635·20-s + 0.602·21-s − 1.97·22-s + 1.41·23-s + 0.146·24-s − 0.387·25-s + 1.57·26-s − 0.192·27-s − 0.847·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.539370662$
$L(\frac12)$  $\approx$  $1.539370662$
$L(\frac{3}{2})$   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 \( 1 - 1.90T + 2T^{2} \)
5 \( 1 + 1.74T + 5T^{2} \)
7 \( 1 + 2.76T + 7T^{2} \)
11 \( 1 + 4.86T + 11T^{2} \)
13 \( 1 - 4.21T + 13T^{2} \)
19 \( 1 + 2.06T + 19T^{2} \)
23 \( 1 - 6.78T + 23T^{2} \)
29 \( 1 - 1.05T + 29T^{2} \)
31 \( 1 + 1.35T + 31T^{2} \)
37 \( 1 - 1.37T + 37T^{2} \)
41 \( 1 - 2.38T + 41T^{2} \)
43 \( 1 - 2.48T + 43T^{2} \)
47 \( 1 - 12.3T + 47T^{2} \)
53 \( 1 - 9.48T + 53T^{2} \)
59 \( 1 - 3.60T + 59T^{2} \)
61 \( 1 - 0.816T + 61T^{2} \)
67 \( 1 - 13.5T + 67T^{2} \)
71 \( 1 + 5.74T + 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
83 \( 1 + 8.93T + 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−8.418298832101021768656461611244, −7.36139385370654881055633879400, −6.81300328950957663380134343883, −5.90291383050289970207917050962, −5.56116286740564808015916282513, −4.59311251727870628750290220724, −3.95116412100566344621011103319, −3.22797718432823634544786820653, −2.45928883907839541960558511458, −0.56633668115182396845649715471, 0.56633668115182396845649715471, 2.45928883907839541960558511458, 3.22797718432823634544786820653, 3.95116412100566344621011103319, 4.59311251727870628750290220724, 5.56116286740564808015916282513, 5.90291383050289970207917050962, 6.81300328950957663380134343883, 7.36139385370654881055633879400, 8.418298832101021768656461611244

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