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  − 0.249·2-s − 3-s − 1.93·4-s + 0.601·5-s + 0.249·6-s − 2.69·7-s + 0.983·8-s + 9-s − 0.150·10-s − 4.00·11-s + 1.93·12-s + 6.29·13-s + 0.672·14-s − 0.601·15-s + 3.62·16-s − 17-s − 0.249·18-s + 7.87·19-s − 1.16·20-s + 2.69·21-s + 1.00·22-s − 0.0277·23-s − 0.983·24-s − 4.63·25-s − 1.57·26-s − 27-s + 5.21·28-s + ⋯
L(s)  = 1  − 0.176·2-s − 0.577·3-s − 0.968·4-s + 0.269·5-s + 0.101·6-s − 1.01·7-s + 0.347·8-s + 0.333·9-s − 0.0475·10-s − 1.20·11-s + 0.559·12-s + 1.74·13-s + 0.179·14-s − 0.155·15-s + 0.907·16-s − 0.242·17-s − 0.0588·18-s + 1.80·19-s − 0.260·20-s + 0.587·21-s + 0.213·22-s − 0.00579·23-s − 0.200·24-s − 0.927·25-s − 0.308·26-s − 0.192·27-s + 0.985·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7239754989$
$L(\frac12)$  $\approx$  $0.7239754989$
$L(\frac{3}{2})$   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 \( 1 + 0.249T + 2T^{2} \)
5 \( 1 - 0.601T + 5T^{2} \)
7 \( 1 + 2.69T + 7T^{2} \)
11 \( 1 + 4.00T + 11T^{2} \)
13 \( 1 - 6.29T + 13T^{2} \)
19 \( 1 - 7.87T + 19T^{2} \)
23 \( 1 + 0.0277T + 23T^{2} \)
29 \( 1 + 7.65T + 29T^{2} \)
31 \( 1 - 3.60T + 31T^{2} \)
37 \( 1 + 3.13T + 37T^{2} \)
41 \( 1 + 8.77T + 41T^{2} \)
43 \( 1 + 5.84T + 43T^{2} \)
47 \( 1 + 6.01T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 0.767T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 - 3.73T + 71T^{2} \)
73 \( 1 - 3.98T + 73T^{2} \)
83 \( 1 - 6.87T + 83T^{2} \)
89 \( 1 + 1.23T + 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−8.391201021549054602848323058540, −7.87906605793611504644518538034, −6.90516998556910299658779000897, −6.08970289763510214235854050777, −5.45810964333742368448844377718, −4.91598675825058483251279758339, −3.61821477176592887138846925865, −3.33931337269066298071087834878, −1.70894640802821754485663986439, −0.52604235847985783446426664158, 0.52604235847985783446426664158, 1.70894640802821754485663986439, 3.33931337269066298071087834878, 3.61821477176592887138846925865, 4.91598675825058483251279758339, 5.45810964333742368448844377718, 6.08970289763510214235854050777, 6.90516998556910299658779000897, 7.87906605793611504644518538034, 8.391201021549054602848323058540

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