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  − 1.61·2-s − 3-s + 0.621·4-s + 1.62·5-s + 1.61·6-s + 3.15·7-s + 2.23·8-s + 9-s − 2.62·10-s − 2.23·11-s − 0.621·12-s − 1.49·13-s − 5.11·14-s − 1.62·15-s − 4.85·16-s − 17-s − 1.61·18-s − 3.38·19-s + 1.00·20-s − 3.15·21-s + 3.61·22-s + 7.59·23-s − 2.23·24-s − 2.37·25-s + 2.41·26-s − 27-s + 1.96·28-s + ⋯
L(s)  = 1  − 1.14·2-s − 0.577·3-s + 0.310·4-s + 0.725·5-s + 0.661·6-s + 1.19·7-s + 0.789·8-s + 0.333·9-s − 0.830·10-s − 0.673·11-s − 0.179·12-s − 0.413·13-s − 1.36·14-s − 0.418·15-s − 1.21·16-s − 0.242·17-s − 0.381·18-s − 0.776·19-s + 0.225·20-s − 0.689·21-s + 0.770·22-s + 1.58·23-s − 0.455·24-s − 0.474·25-s + 0.473·26-s − 0.192·27-s + 0.371·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8959914985$
$L(\frac12)$  $\approx$  $0.8959914985$
$L(\frac{3}{2})$   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 \( 1 + 1.61T + 2T^{2} \)
5 \( 1 - 1.62T + 5T^{2} \)
7 \( 1 - 3.15T + 7T^{2} \)
11 \( 1 + 2.23T + 11T^{2} \)
13 \( 1 + 1.49T + 13T^{2} \)
19 \( 1 + 3.38T + 19T^{2} \)
23 \( 1 - 7.59T + 23T^{2} \)
29 \( 1 + 6.67T + 29T^{2} \)
31 \( 1 + 2.84T + 31T^{2} \)
37 \( 1 - 7.70T + 37T^{2} \)
41 \( 1 - 1.42T + 41T^{2} \)
43 \( 1 - 8.92T + 43T^{2} \)
47 \( 1 + 8.52T + 47T^{2} \)
53 \( 1 - 7.27T + 53T^{2} \)
59 \( 1 - 1.82T + 59T^{2} \)
61 \( 1 - 1.27T + 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 13.7T + 73T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 - 3.88T + 97T^{2} \)
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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

−8.485266678919941794948036413293, −7.74577020595522599039624050124, −7.28879016628730793239122958574, −6.32873933239224054726158375174, −5.35574957794584454943266645113, −4.92063100976056147467369579492, −4.05444731715266286464066647692, −2.40135580234800064849155376442, −1.75381687133843549934283851821, −0.68134204419067878525433273602, 0.68134204419067878525433273602, 1.75381687133843549934283851821, 2.40135580234800064849155376442, 4.05444731715266286464066647692, 4.92063100976056147467369579492, 5.35574957794584454943266645113, 6.32873933239224054726158375174, 7.28879016628730793239122958574, 7.74577020595522599039624050124, 8.485266678919941794948036413293

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