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  − 0.0717·2-s − 3-s − 1.99·4-s − 1.92·5-s + 0.0717·6-s + 3.89·7-s + 0.286·8-s + 9-s + 0.137·10-s + 0.330·11-s + 1.99·12-s + 1.95·13-s − 0.279·14-s + 1.92·15-s + 3.96·16-s − 17-s − 0.0717·18-s + 7.45·19-s + 3.83·20-s − 3.89·21-s − 0.0236·22-s − 0.344·23-s − 0.286·24-s − 1.29·25-s − 0.140·26-s − 27-s − 7.76·28-s + ⋯
L(s)  = 1  − 0.0507·2-s − 0.577·3-s − 0.997·4-s − 0.860·5-s + 0.0292·6-s + 1.47·7-s + 0.101·8-s + 0.333·9-s + 0.0436·10-s + 0.0995·11-s + 0.575·12-s + 0.542·13-s − 0.0746·14-s + 0.496·15-s + 0.992·16-s − 0.242·17-s − 0.0169·18-s + 1.71·19-s + 0.858·20-s − 0.849·21-s − 0.00505·22-s − 0.0718·23-s − 0.0584·24-s − 0.259·25-s − 0.0275·26-s − 0.192·27-s − 1.46·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.203170637$
$L(\frac12)$  $\approx$  $1.203170637$
$L(\frac{3}{2})$   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 \( 1 + 0.0717T + 2T^{2} \)
5 \( 1 + 1.92T + 5T^{2} \)
7 \( 1 - 3.89T + 7T^{2} \)
11 \( 1 - 0.330T + 11T^{2} \)
13 \( 1 - 1.95T + 13T^{2} \)
19 \( 1 - 7.45T + 19T^{2} \)
23 \( 1 + 0.344T + 23T^{2} \)
29 \( 1 + 3.99T + 29T^{2} \)
31 \( 1 - 6.60T + 31T^{2} \)
37 \( 1 + 0.669T + 37T^{2} \)
41 \( 1 + 3.05T + 41T^{2} \)
43 \( 1 - 4.13T + 43T^{2} \)
47 \( 1 - 3.35T + 47T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 + 14.3T + 59T^{2} \)
61 \( 1 - 9.85T + 61T^{2} \)
67 \( 1 + 7.02T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + 8.01T + 97T^{2} \)
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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

−8.466490649307748596057826243391, −7.61349604896222023274237739894, −7.38833878712373201970141703437, −5.96785290447495704884265685182, −5.37058885357714642777127013663, −4.58815938040811587331490738159, −4.14748082192480787886235201081, −3.20545539676812796442665590880, −1.59112939422867568310140697007, −0.71463169451144713378904167196, 0.71463169451144713378904167196, 1.59112939422867568310140697007, 3.20545539676812796442665590880, 4.14748082192480787886235201081, 4.58815938040811587331490738159, 5.37058885357714642777127013663, 5.96785290447495704884265685182, 7.38833878712373201970141703437, 7.61349604896222023274237739894, 8.466490649307748596057826243391

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