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  − 0.103·2-s + 3-s − 1.98·4-s − 2.25·5-s − 0.103·6-s + 4.11·7-s + 0.412·8-s + 9-s + 0.233·10-s + 3.41·11-s − 1.98·12-s + 3.88·13-s − 0.424·14-s − 2.25·15-s + 3.93·16-s + 17-s − 0.103·18-s − 2.55·19-s + 4.48·20-s + 4.11·21-s − 0.353·22-s + 2.82·23-s + 0.412·24-s + 0.0923·25-s − 0.401·26-s + 27-s − 8.17·28-s + ⋯
L(s)  = 1  − 0.0730·2-s + 0.577·3-s − 0.994·4-s − 1.00·5-s − 0.0421·6-s + 1.55·7-s + 0.145·8-s + 0.333·9-s + 0.0737·10-s + 1.02·11-s − 0.574·12-s + 1.07·13-s − 0.113·14-s − 0.582·15-s + 0.984·16-s + 0.242·17-s − 0.0243·18-s − 0.586·19-s + 1.00·20-s + 0.897·21-s − 0.0752·22-s + 0.589·23-s + 0.0841·24-s + 0.0184·25-s − 0.0786·26-s + 0.192·27-s − 1.54·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.110434847$
$L(\frac12)$  $\approx$  $2.110434847$
$L(\frac{3}{2})$   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 \( 1 + 0.103T + 2T^{2} \)
5 \( 1 + 2.25T + 5T^{2} \)
7 \( 1 - 4.11T + 7T^{2} \)
11 \( 1 - 3.41T + 11T^{2} \)
13 \( 1 - 3.88T + 13T^{2} \)
19 \( 1 + 2.55T + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 - 9.03T + 29T^{2} \)
31 \( 1 - 2.07T + 31T^{2} \)
37 \( 1 - 0.222T + 37T^{2} \)
41 \( 1 + 9.66T + 41T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 - 0.936T + 47T^{2} \)
53 \( 1 + 1.47T + 53T^{2} \)
59 \( 1 - 8.03T + 59T^{2} \)
61 \( 1 - 1.23T + 61T^{2} \)
67 \( 1 + 5.75T + 67T^{2} \)
71 \( 1 - 14.6T + 71T^{2} \)
73 \( 1 + 9.54T + 73T^{2} \)
83 \( 1 - 7.96T + 83T^{2} \)
89 \( 1 - 0.331T + 89T^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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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

−8.416573934092287313410647397669, −8.149320257112475149941774597414, −7.18513551942727134899896724552, −6.32856217893966083930727775281, −5.11196388958246587072339423577, −4.55536077631780489526528568021, −3.91623326885732624894126512550, −3.27591555514979695880352230487, −1.71314437977392451508145501802, −0.911101847982377059564661825922, 0.911101847982377059564661825922, 1.71314437977392451508145501802, 3.27591555514979695880352230487, 3.91623326885732624894126512550, 4.55536077631780489526528568021, 5.11196388958246587072339423577, 6.32856217893966083930727775281, 7.18513551942727134899896724552, 8.149320257112475149941774597414, 8.416573934092287313410647397669

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