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  + 1.06·2-s − 3-s − 0.856·4-s − 2.68·5-s − 1.06·6-s + 1.02·7-s − 3.05·8-s + 9-s − 2.87·10-s − 5.48·11-s + 0.856·12-s − 4.05·13-s + 1.09·14-s + 2.68·15-s − 1.55·16-s − 17-s + 1.06·18-s + 5.30·19-s + 2.30·20-s − 1.02·21-s − 5.86·22-s − 7.24·23-s + 3.05·24-s + 2.23·25-s − 4.33·26-s − 27-s − 0.880·28-s + ⋯
L(s)  = 1  + 0.756·2-s − 0.577·3-s − 0.428·4-s − 1.20·5-s − 0.436·6-s + 0.388·7-s − 1.07·8-s + 0.333·9-s − 0.909·10-s − 1.65·11-s + 0.247·12-s − 1.12·13-s + 0.293·14-s + 0.694·15-s − 0.388·16-s − 0.242·17-s + 0.252·18-s + 1.21·19-s + 0.515·20-s − 0.224·21-s − 1.25·22-s − 1.51·23-s + 0.623·24-s + 0.447·25-s − 0.850·26-s − 0.192·27-s − 0.166·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.3209465584$
$L(\frac12)$  $\approx$  $0.3209465584$
$L(\frac{3}{2})$   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 \( 1 - 1.06T + 2T^{2} \)
5 \( 1 + 2.68T + 5T^{2} \)
7 \( 1 - 1.02T + 7T^{2} \)
11 \( 1 + 5.48T + 11T^{2} \)
13 \( 1 + 4.05T + 13T^{2} \)
19 \( 1 - 5.30T + 19T^{2} \)
23 \( 1 + 7.24T + 23T^{2} \)
29 \( 1 + 7.56T + 29T^{2} \)
31 \( 1 + 10.0T + 31T^{2} \)
37 \( 1 - 0.872T + 37T^{2} \)
41 \( 1 - 2.54T + 41T^{2} \)
43 \( 1 - 7.71T + 43T^{2} \)
47 \( 1 - 1.60T + 47T^{2} \)
53 \( 1 - 3.09T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 + 4.20T + 61T^{2} \)
67 \( 1 - 9.58T + 67T^{2} \)
71 \( 1 + 8.78T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + 9.31T + 89T^{2} \)
97 \( 1 + 3.52T + 97T^{2} \)
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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

−8.062728301862569992353126133356, −7.73157720778744043413497926456, −7.15558201159195541234779745142, −5.76621736952557263717849595628, −5.41407499411331421251449987962, −4.69357012046104157489950992190, −4.02847316411165036560045085788, −3.24707924008430902607106196612, −2.17575984926719871694473849638, −0.27967353701640543691449171198, 0.27967353701640543691449171198, 2.17575984926719871694473849638, 3.24707924008430902607106196612, 4.02847316411165036560045085788, 4.69357012046104157489950992190, 5.41407499411331421251449987962, 5.76621736952557263717849595628, 7.15558201159195541234779745142, 7.73157720778744043413497926456, 8.062728301862569992353126133356

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