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  − 0.626·2-s + 3-s − 1.60·4-s − 1.73·5-s − 0.626·6-s − 3.52·7-s + 2.25·8-s + 9-s + 1.08·10-s − 5.10·11-s − 1.60·12-s − 6.25·13-s + 2.21·14-s − 1.73·15-s + 1.79·16-s + 17-s − 0.626·18-s − 1.32·19-s + 2.79·20-s − 3.52·21-s + 3.19·22-s − 4.67·23-s + 2.25·24-s − 1.98·25-s + 3.91·26-s + 27-s + 5.67·28-s + ⋯
L(s)  = 1  − 0.442·2-s + 0.577·3-s − 0.803·4-s − 0.776·5-s − 0.255·6-s − 1.33·7-s + 0.798·8-s + 0.333·9-s + 0.343·10-s − 1.53·11-s − 0.464·12-s − 1.73·13-s + 0.590·14-s − 0.448·15-s + 0.449·16-s + 0.242·17-s − 0.147·18-s − 0.303·19-s + 0.624·20-s − 0.770·21-s + 0.682·22-s − 0.974·23-s + 0.461·24-s − 0.397·25-s + 0.768·26-s + 0.192·27-s + 1.07·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.04931329400$
$L(\frac12)$  $\approx$  $0.04931329400$
$L(\frac{3}{2})$   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 \( 1 + 0.626T + 2T^{2} \)
5 \( 1 + 1.73T + 5T^{2} \)
7 \( 1 + 3.52T + 7T^{2} \)
11 \( 1 + 5.10T + 11T^{2} \)
13 \( 1 + 6.25T + 13T^{2} \)
19 \( 1 + 1.32T + 19T^{2} \)
23 \( 1 + 4.67T + 23T^{2} \)
29 \( 1 + 1.94T + 29T^{2} \)
31 \( 1 + 2.74T + 31T^{2} \)
37 \( 1 - 2.56T + 37T^{2} \)
41 \( 1 + 5.78T + 41T^{2} \)
43 \( 1 - 2.16T + 43T^{2} \)
47 \( 1 + 12.8T + 47T^{2} \)
53 \( 1 + 4.09T + 53T^{2} \)
59 \( 1 + 5.16T + 59T^{2} \)
61 \( 1 + 9.98T + 61T^{2} \)
67 \( 1 + 2.06T + 67T^{2} \)
71 \( 1 - 7.20T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 + 14.4T + 89T^{2} \)
97 \( 1 + 4.30T + 97T^{2} \)
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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

−8.297489366388059566888973913439, −7.76125100391119354633920398615, −7.42893686489713294921608594763, −6.36552014927312977941356356035, −5.26969703490787741654221048123, −4.64854146173712777165519200973, −3.71542140061498311194867805920, −3.04130833612495432027441425706, −2.08227058711084845438935601374, −0.12285119499056015249624317130, 0.12285119499056015249624317130, 2.08227058711084845438935601374, 3.04130833612495432027441425706, 3.71542140061498311194867805920, 4.64854146173712777165519200973, 5.26969703490787741654221048123, 6.36552014927312977941356356035, 7.42893686489713294921608594763, 7.76125100391119354633920398615, 8.297489366388059566888973913439

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