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  + 0.403·2-s − 3-s − 1.83·4-s + 2.23·5-s − 0.403·6-s − 0.778·7-s − 1.54·8-s + 9-s + 0.901·10-s + 0.498·11-s + 1.83·12-s − 0.674·13-s − 0.313·14-s − 2.23·15-s + 3.05·16-s − 17-s + 0.403·18-s − 7.73·19-s − 4.10·20-s + 0.778·21-s + 0.200·22-s + 2.53·23-s + 1.54·24-s + 0.00242·25-s − 0.271·26-s − 27-s + 1.43·28-s + ⋯
L(s)  = 1  + 0.285·2-s − 0.577·3-s − 0.918·4-s + 1.00·5-s − 0.164·6-s − 0.294·7-s − 0.547·8-s + 0.333·9-s + 0.285·10-s + 0.150·11-s + 0.530·12-s − 0.186·13-s − 0.0839·14-s − 0.577·15-s + 0.762·16-s − 0.242·17-s + 0.0950·18-s − 1.77·19-s − 0.918·20-s + 0.169·21-s + 0.0428·22-s + 0.528·23-s + 0.315·24-s + 0.000484·25-s − 0.0533·26-s − 0.192·27-s + 0.270·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.343178739$
$L(\frac12)$  $\approx$  $1.343178739$
$L(\frac{3}{2})$   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 \( 1 - 0.403T + 2T^{2} \)
5 \( 1 - 2.23T + 5T^{2} \)
7 \( 1 + 0.778T + 7T^{2} \)
11 \( 1 - 0.498T + 11T^{2} \)
13 \( 1 + 0.674T + 13T^{2} \)
19 \( 1 + 7.73T + 19T^{2} \)
23 \( 1 - 2.53T + 23T^{2} \)
29 \( 1 + 1.10T + 29T^{2} \)
31 \( 1 - 8.74T + 31T^{2} \)
37 \( 1 - 2.84T + 37T^{2} \)
41 \( 1 - 5.58T + 41T^{2} \)
43 \( 1 + 1.12T + 43T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 + 1.21T + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 + 8.15T + 61T^{2} \)
67 \( 1 - 1.40T + 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 - 8.98T + 73T^{2} \)
83 \( 1 - 12.9T + 83T^{2} \)
89 \( 1 + 7.67T + 89T^{2} \)
97 \( 1 + 3.48T + 97T^{2} \)
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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

−8.556581133572175924375832405400, −7.77663649956107535906805330492, −6.55374772815980857867213118109, −6.22067764688798107054110337095, −5.50007309382175470130590207475, −4.64376843949899049097903946907, −4.16148542397104716844951080995, −2.97156373827142145085281389393, −1.95953420974712789721530433335, −0.65312991770969700403449571202, 0.65312991770969700403449571202, 1.95953420974712789721530433335, 2.97156373827142145085281389393, 4.16148542397104716844951080995, 4.64376843949899049097903946907, 5.50007309382175470130590207475, 6.22067764688798107054110337095, 6.55374772815980857867213118109, 7.77663649956107535906805330492, 8.556581133572175924375832405400

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