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  − 2.16·2-s − 3-s + 2.69·4-s + 2.47·5-s + 2.16·6-s + 5.12·7-s − 1.51·8-s + 9-s − 5.36·10-s + 2.42·11-s − 2.69·12-s + 7.04·13-s − 11.0·14-s − 2.47·15-s − 2.11·16-s − 17-s − 2.16·18-s + 2.52·19-s + 6.67·20-s − 5.12·21-s − 5.26·22-s − 3.87·23-s + 1.51·24-s + 1.12·25-s − 15.2·26-s − 27-s + 13.8·28-s + ⋯
L(s)  = 1  − 1.53·2-s − 0.577·3-s + 1.34·4-s + 1.10·5-s + 0.884·6-s + 1.93·7-s − 0.535·8-s + 0.333·9-s − 1.69·10-s + 0.732·11-s − 0.778·12-s + 1.95·13-s − 2.96·14-s − 0.639·15-s − 0.528·16-s − 0.242·17-s − 0.510·18-s + 0.579·19-s + 1.49·20-s − 1.11·21-s − 1.12·22-s − 0.808·23-s + 0.309·24-s + 0.225·25-s − 2.99·26-s − 0.192·27-s + 2.61·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.484823589$
$L(\frac12)$  $\approx$  $1.484823589$
$L(\frac{3}{2})$   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 \( 1 + 2.16T + 2T^{2} \)
5 \( 1 - 2.47T + 5T^{2} \)
7 \( 1 - 5.12T + 7T^{2} \)
11 \( 1 - 2.42T + 11T^{2} \)
13 \( 1 - 7.04T + 13T^{2} \)
19 \( 1 - 2.52T + 19T^{2} \)
23 \( 1 + 3.87T + 23T^{2} \)
29 \( 1 + 0.167T + 29T^{2} \)
31 \( 1 - 5.21T + 31T^{2} \)
37 \( 1 - 2.43T + 37T^{2} \)
41 \( 1 + 1.78T + 41T^{2} \)
43 \( 1 + 0.311T + 43T^{2} \)
47 \( 1 - 3.61T + 47T^{2} \)
53 \( 1 - 5.51T + 53T^{2} \)
59 \( 1 - 3.98T + 59T^{2} \)
61 \( 1 + 4.00T + 61T^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 + 6.09T + 71T^{2} \)
73 \( 1 - 7.55T + 73T^{2} \)
83 \( 1 - 2.53T + 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 - 7.77T + 97T^{2} \)
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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

−8.555830339368335223286899820563, −7.957000682328365212892212679048, −7.17264325933059238458480520464, −6.22243300426498644892823415481, −5.80145465340656283099824654448, −4.78733667203476154293594507883, −3.96930675949385976205111057619, −2.26966953197858681169163789020, −1.43324735832291164555366078161, −1.12359803756067426329942289062, 1.12359803756067426329942289062, 1.43324735832291164555366078161, 2.26966953197858681169163789020, 3.96930675949385976205111057619, 4.78733667203476154293594507883, 5.80145465340656283099824654448, 6.22243300426498644892823415481, 7.17264325933059238458480520464, 7.957000682328365212892212679048, 8.555830339368335223286899820563

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