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  + 1.26·2-s − 3-s − 0.401·4-s + 1.64·5-s − 1.26·6-s + 5.15·7-s − 3.03·8-s + 9-s + 2.07·10-s − 5.89·11-s + 0.401·12-s + 2.01·13-s + 6.51·14-s − 1.64·15-s − 3.03·16-s − 17-s + 1.26·18-s + 1.52·19-s − 0.658·20-s − 5.15·21-s − 7.44·22-s + 0.245·23-s + 3.03·24-s − 2.30·25-s + 2.54·26-s − 27-s − 2.06·28-s + ⋯
L(s)  = 1  + 0.894·2-s − 0.577·3-s − 0.200·4-s + 0.733·5-s − 0.516·6-s + 1.94·7-s − 1.07·8-s + 0.333·9-s + 0.655·10-s − 1.77·11-s + 0.115·12-s + 0.558·13-s + 1.74·14-s − 0.423·15-s − 0.759·16-s − 0.242·17-s + 0.298·18-s + 0.350·19-s − 0.147·20-s − 1.12·21-s − 1.58·22-s + 0.0512·23-s + 0.619·24-s − 0.461·25-s + 0.499·26-s − 0.192·27-s − 0.391·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.851102255$
$L(\frac12)$  $\approx$  $2.851102255$
$L(\frac{3}{2})$   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 \( 1 - 1.26T + 2T^{2} \)
5 \( 1 - 1.64T + 5T^{2} \)
7 \( 1 - 5.15T + 7T^{2} \)
11 \( 1 + 5.89T + 11T^{2} \)
13 \( 1 - 2.01T + 13T^{2} \)
19 \( 1 - 1.52T + 19T^{2} \)
23 \( 1 - 0.245T + 23T^{2} \)
29 \( 1 - 9.74T + 29T^{2} \)
31 \( 1 - 7.83T + 31T^{2} \)
37 \( 1 - 3.04T + 37T^{2} \)
41 \( 1 + 6.30T + 41T^{2} \)
43 \( 1 - 2.84T + 43T^{2} \)
47 \( 1 - 5.05T + 47T^{2} \)
53 \( 1 + 7.85T + 53T^{2} \)
59 \( 1 - 5.24T + 59T^{2} \)
61 \( 1 - 0.259T + 61T^{2} \)
67 \( 1 - 2.83T + 67T^{2} \)
71 \( 1 + 1.86T + 71T^{2} \)
73 \( 1 + 0.762T + 73T^{2} \)
83 \( 1 + 0.429T + 83T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−8.159196133847917721846137098162, −7.972199409073044966746263018160, −6.71636261160567581993177527950, −5.84332382882555214220308449039, −5.33470764556001137402572119061, −4.79542244486411584420524481065, −4.30249888379734053561617294924, −2.92386816310451207381751858155, −2.11309889278927445416723732994, −0.898969197176852867278467789047, 0.898969197176852867278467789047, 2.11309889278927445416723732994, 2.92386816310451207381751858155, 4.30249888379734053561617294924, 4.79542244486411584420524481065, 5.33470764556001137402572119061, 5.84332382882555214220308449039, 6.71636261160567581993177527950, 7.972199409073044966746263018160, 8.159196133847917721846137098162

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