Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 79 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.17·2-s − 3-s − 0.621·4-s + 3.86·5-s + 1.17·6-s + 3.06·7-s + 3.07·8-s + 9-s − 4.54·10-s + 1.94·11-s + 0.621·12-s − 4.31·13-s − 3.59·14-s − 3.86·15-s − 2.37·16-s − 17-s − 1.17·18-s − 5.08·19-s − 2.40·20-s − 3.06·21-s − 2.28·22-s − 2.65·23-s − 3.07·24-s + 9.96·25-s + 5.06·26-s − 27-s − 1.90·28-s + ⋯
L(s)  = 1  − 0.830·2-s − 0.577·3-s − 0.310·4-s + 1.73·5-s + 0.479·6-s + 1.15·7-s + 1.08·8-s + 0.333·9-s − 1.43·10-s + 0.586·11-s + 0.179·12-s − 1.19·13-s − 0.960·14-s − 0.998·15-s − 0.592·16-s − 0.242·17-s − 0.276·18-s − 1.16·19-s − 0.537·20-s − 0.668·21-s − 0.487·22-s − 0.553·23-s − 0.628·24-s + 1.99·25-s + 0.992·26-s − 0.192·27-s − 0.359·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.404687219$
$L(\frac12)$  $\approx$  $1.404687219$
$L(\frac{3}{2})$   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 \( 1 + 1.17T + 2T^{2} \)
5 \( 1 - 3.86T + 5T^{2} \)
7 \( 1 - 3.06T + 7T^{2} \)
11 \( 1 - 1.94T + 11T^{2} \)
13 \( 1 + 4.31T + 13T^{2} \)
19 \( 1 + 5.08T + 19T^{2} \)
23 \( 1 + 2.65T + 23T^{2} \)
29 \( 1 - 5.36T + 29T^{2} \)
31 \( 1 - 0.729T + 31T^{2} \)
37 \( 1 + 6.55T + 37T^{2} \)
41 \( 1 - 8.81T + 41T^{2} \)
43 \( 1 - 5.21T + 43T^{2} \)
47 \( 1 - 5.95T + 47T^{2} \)
53 \( 1 + 1.03T + 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 - 5.20T + 61T^{2} \)
67 \( 1 + 7.84T + 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 + 14.0T + 73T^{2} \)
83 \( 1 - 1.61T + 83T^{2} \)
89 \( 1 + 2.37T + 89T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.740367831677847001318702007268, −7.79201615571536672117304088799, −7.01400555092410892018862665811, −6.23798458042492428231177148118, −5.45488400851186438701920103347, −4.79002357323332216443079056403, −4.22426447647384020320744069276, −2.34473055540631890587868020794, −1.82937221207146112348504451789, −0.832315915935347838629209976950, 0.832315915935347838629209976950, 1.82937221207146112348504451789, 2.34473055540631890587868020794, 4.22426447647384020320744069276, 4.79002357323332216443079056403, 5.45488400851186438701920103347, 6.23798458042492428231177148118, 7.01400555092410892018862665811, 7.79201615571536672117304088799, 8.740367831677847001318702007268

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