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  + 0.437·2-s − 3-s − 1.80·4-s − 1.50·5-s − 0.437·6-s − 4.33·7-s − 1.66·8-s + 9-s − 0.658·10-s + 3.26·11-s + 1.80·12-s − 1.64·13-s − 1.89·14-s + 1.50·15-s + 2.88·16-s − 17-s + 0.437·18-s − 5.97·19-s + 2.71·20-s + 4.33·21-s + 1.42·22-s − 5.62·23-s + 1.66·24-s − 2.74·25-s − 0.720·26-s − 27-s + 7.84·28-s + ⋯
L(s)  = 1  + 0.309·2-s − 0.577·3-s − 0.904·4-s − 0.672·5-s − 0.178·6-s − 1.63·7-s − 0.589·8-s + 0.333·9-s − 0.208·10-s + 0.983·11-s + 0.521·12-s − 0.456·13-s − 0.507·14-s + 0.388·15-s + 0.721·16-s − 0.242·17-s + 0.103·18-s − 1.37·19-s + 0.607·20-s + 0.946·21-s + 0.304·22-s − 1.17·23-s + 0.340·24-s − 0.548·25-s − 0.141·26-s − 0.192·27-s + 1.48·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.02891164422$
$L(\frac12)$  $\approx$  $0.02891164422$
$L(\frac{3}{2})$   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 \( 1 - 0.437T + 2T^{2} \)
5 \( 1 + 1.50T + 5T^{2} \)
7 \( 1 + 4.33T + 7T^{2} \)
11 \( 1 - 3.26T + 11T^{2} \)
13 \( 1 + 1.64T + 13T^{2} \)
19 \( 1 + 5.97T + 19T^{2} \)
23 \( 1 + 5.62T + 23T^{2} \)
29 \( 1 + 9.77T + 29T^{2} \)
31 \( 1 + 9.36T + 31T^{2} \)
37 \( 1 + 4.38T + 37T^{2} \)
41 \( 1 + 1.29T + 41T^{2} \)
43 \( 1 + 3.12T + 43T^{2} \)
47 \( 1 + 3.57T + 47T^{2} \)
53 \( 1 + 1.24T + 53T^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 + 4.49T + 61T^{2} \)
67 \( 1 + 8.36T + 67T^{2} \)
71 \( 1 - 2.73T + 71T^{2} \)
73 \( 1 + 15.4T + 73T^{2} \)
83 \( 1 - 2.32T + 83T^{2} \)
89 \( 1 - 1.70T + 89T^{2} \)
97 \( 1 - 1.17T + 97T^{2} \)
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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

−8.628779848869426402992686338002, −7.57019449790180053837120756332, −6.86316971263331974550051004282, −6.10139084599089535815441915985, −5.61585147596111185837949116429, −4.47903679439802706316099158412, −3.80248549855949514738390626712, −3.48897519000851882723285473111, −1.94440741218952136472379442733, −0.091045633880287094613261414961, 0.091045633880287094613261414961, 1.94440741218952136472379442733, 3.48897519000851882723285473111, 3.80248549855949514738390626712, 4.47903679439802706316099158412, 5.61585147596111185837949116429, 6.10139084599089535815441915985, 6.86316971263331974550051004282, 7.57019449790180053837120756332, 8.628779848869426402992686338002

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