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  − 2.57·2-s − 3-s + 4.64·4-s + 2.80·5-s + 2.57·6-s − 2.75·7-s − 6.80·8-s + 9-s − 7.22·10-s + 5.08·11-s − 4.64·12-s + 1.22·13-s + 7.09·14-s − 2.80·15-s + 8.26·16-s − 17-s − 2.57·18-s + 4.27·19-s + 13.0·20-s + 2.75·21-s − 13.1·22-s + 3.53·23-s + 6.80·24-s + 2.85·25-s − 3.16·26-s − 27-s − 12.7·28-s + ⋯
L(s)  = 1  − 1.82·2-s − 0.577·3-s + 2.32·4-s + 1.25·5-s + 1.05·6-s − 1.04·7-s − 2.40·8-s + 0.333·9-s − 2.28·10-s + 1.53·11-s − 1.33·12-s + 0.340·13-s + 1.89·14-s − 0.723·15-s + 2.06·16-s − 0.242·17-s − 0.607·18-s + 0.980·19-s + 2.90·20-s + 0.600·21-s − 2.79·22-s + 0.737·23-s + 1.38·24-s + 0.570·25-s − 0.620·26-s − 0.192·27-s − 2.41·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9252288143$
$L(\frac12)$  $\approx$  $0.9252288143$
$L(\frac{3}{2})$   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 \( 1 + 2.57T + 2T^{2} \)
5 \( 1 - 2.80T + 5T^{2} \)
7 \( 1 + 2.75T + 7T^{2} \)
11 \( 1 - 5.08T + 11T^{2} \)
13 \( 1 - 1.22T + 13T^{2} \)
19 \( 1 - 4.27T + 19T^{2} \)
23 \( 1 - 3.53T + 23T^{2} \)
29 \( 1 - 5.81T + 29T^{2} \)
31 \( 1 - 6.37T + 31T^{2} \)
37 \( 1 - 5.23T + 37T^{2} \)
41 \( 1 + 0.440T + 41T^{2} \)
43 \( 1 + 1.81T + 43T^{2} \)
47 \( 1 + 6.02T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + 4.07T + 59T^{2} \)
61 \( 1 - 7.50T + 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 - 5.18T + 71T^{2} \)
73 \( 1 - 7.17T + 73T^{2} \)
83 \( 1 - 7.95T + 83T^{2} \)
89 \( 1 + 9.67T + 89T^{2} \)
97 \( 1 - 6.88T + 97T^{2} \)
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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

−8.726173049593258590602284191945, −7.83575367930990218481936968709, −6.70893639652527871801176774871, −6.54787189125113776035989690033, −6.06406827472061185400741944682, −4.92144942038139086084806534387, −3.48329698673758623195335474534, −2.55464827478392132054698629322, −1.46357289611468658230159906155, −0.826986920900155936117324206955, 0.826986920900155936117324206955, 1.46357289611468658230159906155, 2.55464827478392132054698629322, 3.48329698673758623195335474534, 4.92144942038139086084806534387, 6.06406827472061185400741944682, 6.54787189125113776035989690033, 6.70893639652527871801176774871, 7.83575367930990218481936968709, 8.726173049593258590602284191945

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