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  − 1.49·2-s + 3-s + 0.233·4-s − 1.69·5-s − 1.49·6-s + 0.268·7-s + 2.63·8-s + 9-s + 2.53·10-s + 0.913·11-s + 0.233·12-s + 6.63·13-s − 0.401·14-s − 1.69·15-s − 4.41·16-s + 17-s − 1.49·18-s + 0.913·19-s − 0.396·20-s + 0.268·21-s − 1.36·22-s + 4.49·23-s + 2.63·24-s − 2.12·25-s − 9.91·26-s + 27-s + 0.0627·28-s + ⋯
L(s)  = 1  − 1.05·2-s + 0.577·3-s + 0.116·4-s − 0.758·5-s − 0.610·6-s + 0.101·7-s + 0.933·8-s + 0.333·9-s + 0.801·10-s + 0.275·11-s + 0.0674·12-s + 1.83·13-s − 0.107·14-s − 0.437·15-s − 1.10·16-s + 0.242·17-s − 0.352·18-s + 0.209·19-s − 0.0885·20-s + 0.0586·21-s − 0.291·22-s + 0.936·23-s + 0.538·24-s − 0.425·25-s − 1.94·26-s + 0.192·27-s + 0.0118·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.267318061$
$L(\frac12)$  $\approx$  $1.267318061$
$L(\frac{3}{2})$   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 \( 1 + 1.49T + 2T^{2} \)
5 \( 1 + 1.69T + 5T^{2} \)
7 \( 1 - 0.268T + 7T^{2} \)
11 \( 1 - 0.913T + 11T^{2} \)
13 \( 1 - 6.63T + 13T^{2} \)
19 \( 1 - 0.913T + 19T^{2} \)
23 \( 1 - 4.49T + 23T^{2} \)
29 \( 1 - 5.79T + 29T^{2} \)
31 \( 1 + 6.13T + 31T^{2} \)
37 \( 1 - 0.597T + 37T^{2} \)
41 \( 1 - 6.73T + 41T^{2} \)
43 \( 1 - 6.30T + 43T^{2} \)
47 \( 1 - 8.20T + 47T^{2} \)
53 \( 1 - 7.64T + 53T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 - 0.904T + 61T^{2} \)
67 \( 1 + 5.93T + 67T^{2} \)
71 \( 1 - 3.32T + 71T^{2} \)
73 \( 1 + 0.542T + 73T^{2} \)
83 \( 1 + 3.99T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + 16.5T + 97T^{2} \)
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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

−8.589938618817609329612301167200, −7.83223109991273542038944780713, −7.40613047445351973667591502108, −6.49682895678020327193931596717, −5.53627747310447031659574904855, −4.34954672164139697155877567391, −3.88653947088449973036623511462, −2.94173431887232394066743752807, −1.58541151613233807306223797749, −0.809099136873830277606804673562, 0.809099136873830277606804673562, 1.58541151613233807306223797749, 2.94173431887232394066743752807, 3.88653947088449973036623511462, 4.34954672164139697155877567391, 5.53627747310447031659574904855, 6.49682895678020327193931596717, 7.40613047445351973667591502108, 7.83223109991273542038944780713, 8.589938618817609329612301167200

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