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  − 2.22·2-s + 3-s + 2.92·4-s − 1.63·5-s − 2.22·6-s − 1.90·7-s − 2.06·8-s + 9-s + 3.62·10-s − 4.46·11-s + 2.92·12-s − 1.29·13-s + 4.22·14-s − 1.63·15-s − 1.28·16-s + 17-s − 2.22·18-s − 6.67·19-s − 4.78·20-s − 1.90·21-s + 9.90·22-s + 5.41·23-s − 2.06·24-s − 2.32·25-s + 2.87·26-s + 27-s − 5.57·28-s + ⋯
L(s)  = 1  − 1.56·2-s + 0.577·3-s + 1.46·4-s − 0.730·5-s − 0.906·6-s − 0.719·7-s − 0.728·8-s + 0.333·9-s + 1.14·10-s − 1.34·11-s + 0.845·12-s − 0.359·13-s + 1.12·14-s − 0.421·15-s − 0.320·16-s + 0.242·17-s − 0.523·18-s − 1.53·19-s − 1.07·20-s − 0.415·21-s + 2.11·22-s + 1.12·23-s − 0.420·24-s − 0.465·25-s + 0.563·26-s + 0.192·27-s − 1.05·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.3188760777$
$L(\frac12)$  $\approx$  $0.3188760777$
$L(\frac{3}{2})$   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 \( 1 + 2.22T + 2T^{2} \)
5 \( 1 + 1.63T + 5T^{2} \)
7 \( 1 + 1.90T + 7T^{2} \)
11 \( 1 + 4.46T + 11T^{2} \)
13 \( 1 + 1.29T + 13T^{2} \)
19 \( 1 + 6.67T + 19T^{2} \)
23 \( 1 - 5.41T + 23T^{2} \)
29 \( 1 + 0.385T + 29T^{2} \)
31 \( 1 + 4.16T + 31T^{2} \)
37 \( 1 + 6.13T + 37T^{2} \)
41 \( 1 + 9.88T + 41T^{2} \)
43 \( 1 - 2.00T + 43T^{2} \)
47 \( 1 - 3.04T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 - 3.07T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 + 9.15T + 71T^{2} \)
73 \( 1 + 3.90T + 73T^{2} \)
83 \( 1 - 9.32T + 83T^{2} \)
89 \( 1 + 2.90T + 89T^{2} \)
97 \( 1 + 0.111T + 97T^{2} \)
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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

−8.505640349486320211197160026222, −7.924478502176556930057264280498, −7.19064558278098297134376390566, −6.82601332170379607073018994340, −5.60564662896977345805036946612, −4.58102776858710739880338912117, −3.56344467088249721323820734432, −2.68488769662795557188151768121, −1.87884991867527994529323406409, −0.37678374963793140303504878407, 0.37678374963793140303504878407, 1.87884991867527994529323406409, 2.68488769662795557188151768121, 3.56344467088249721323820734432, 4.58102776858710739880338912117, 5.60564662896977345805036946612, 6.82601332170379607073018994340, 7.19064558278098297134376390566, 7.924478502176556930057264280498, 8.505640349486320211197160026222

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