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  + 1.48·2-s − 3-s + 0.215·4-s + 4.37·5-s − 1.48·6-s − 4.26·7-s − 2.65·8-s + 9-s + 6.50·10-s + 2.03·11-s − 0.215·12-s − 1.63·13-s − 6.34·14-s − 4.37·15-s − 4.38·16-s − 17-s + 1.48·18-s + 6.68·19-s + 0.943·20-s + 4.26·21-s + 3.02·22-s + 0.998·23-s + 2.65·24-s + 14.1·25-s − 2.43·26-s − 27-s − 0.919·28-s + ⋯
L(s)  = 1  + 1.05·2-s − 0.577·3-s + 0.107·4-s + 1.95·5-s − 0.607·6-s − 1.61·7-s − 0.939·8-s + 0.333·9-s + 2.05·10-s + 0.613·11-s − 0.0622·12-s − 0.453·13-s − 1.69·14-s − 1.12·15-s − 1.09·16-s − 0.242·17-s + 0.350·18-s + 1.53·19-s + 0.210·20-s + 0.930·21-s + 0.645·22-s + 0.208·23-s + 0.542·24-s + 2.82·25-s − 0.476·26-s − 0.192·27-s − 0.173·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.898592790$
$L(\frac12)$  $\approx$  $2.898592790$
$L(\frac{3}{2})$   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 \( 1 - 1.48T + 2T^{2} \)
5 \( 1 - 4.37T + 5T^{2} \)
7 \( 1 + 4.26T + 7T^{2} \)
11 \( 1 - 2.03T + 11T^{2} \)
13 \( 1 + 1.63T + 13T^{2} \)
19 \( 1 - 6.68T + 19T^{2} \)
23 \( 1 - 0.998T + 23T^{2} \)
29 \( 1 + 5.55T + 29T^{2} \)
31 \( 1 - 4.45T + 31T^{2} \)
37 \( 1 + 9.01T + 37T^{2} \)
41 \( 1 - 4.65T + 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 - 5.93T + 47T^{2} \)
53 \( 1 + 1.83T + 53T^{2} \)
59 \( 1 - 9.08T + 59T^{2} \)
61 \( 1 - 2.34T + 61T^{2} \)
67 \( 1 - 6.16T + 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 + 3.88T + 73T^{2} \)
83 \( 1 + 13.1T + 83T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 + 11.2T + 97T^{2} \)
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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

−8.852524987695691159695336453303, −7.11276549964155762679788161800, −6.70460560383525273253610057614, −5.88920293315905448361704339775, −5.65690085191399515521086628762, −4.93576998915969512443217556201, −3.85891711859737609415594435940, −3.03283965139200413986652214058, −2.27218229754398757571879489441, −0.857800750055788107386272357740, 0.857800750055788107386272357740, 2.27218229754398757571879489441, 3.03283965139200413986652214058, 3.85891711859737609415594435940, 4.93576998915969512443217556201, 5.65690085191399515521086628762, 5.88920293315905448361704339775, 6.70460560383525273253610057614, 7.11276549964155762679788161800, 8.852524987695691159695336453303

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