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  + 0.308·2-s + 3-s − 1.90·4-s + 2.48·5-s + 0.308·6-s + 3.17·7-s − 1.20·8-s + 9-s + 0.765·10-s − 2.66·11-s − 1.90·12-s + 6.37·13-s + 0.978·14-s + 2.48·15-s + 3.43·16-s + 17-s + 0.308·18-s + 8.52·19-s − 4.72·20-s + 3.17·21-s − 0.822·22-s + 6.90·23-s − 1.20·24-s + 1.16·25-s + 1.96·26-s + 27-s − 6.04·28-s + ⋯
L(s)  = 1  + 0.217·2-s + 0.577·3-s − 0.952·4-s + 1.11·5-s + 0.125·6-s + 1.20·7-s − 0.425·8-s + 0.333·9-s + 0.242·10-s − 0.804·11-s − 0.549·12-s + 1.76·13-s + 0.261·14-s + 0.641·15-s + 0.859·16-s + 0.242·17-s + 0.0726·18-s + 1.95·19-s − 1.05·20-s + 0.692·21-s − 0.175·22-s + 1.43·23-s − 0.245·24-s + 0.232·25-s + 0.385·26-s + 0.192·27-s − 1.14·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.406393155$
$L(\frac12)$  $\approx$  $3.406393155$
$L(\frac{3}{2})$   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 \( 1 - 0.308T + 2T^{2} \)
5 \( 1 - 2.48T + 5T^{2} \)
7 \( 1 - 3.17T + 7T^{2} \)
11 \( 1 + 2.66T + 11T^{2} \)
13 \( 1 - 6.37T + 13T^{2} \)
19 \( 1 - 8.52T + 19T^{2} \)
23 \( 1 - 6.90T + 23T^{2} \)
29 \( 1 + 2.52T + 29T^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 + 1.66T + 41T^{2} \)
43 \( 1 + 1.09T + 43T^{2} \)
47 \( 1 - 3.69T + 47T^{2} \)
53 \( 1 + 0.219T + 53T^{2} \)
59 \( 1 - 10.6T + 59T^{2} \)
61 \( 1 + 4.77T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 - 6.68T + 97T^{2} \)
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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

−8.716681742142359895240910840307, −7.78813090126448336529812530859, −7.19016113118203639915753456358, −5.84873802132001555601248972769, −5.37568046673171427264159710817, −4.91479620341529252830049884217, −3.69112939839868111288315577398, −3.16201413700836099058924049943, −1.81078263166900856976502893480, −1.14185815515325249597961012458, 1.14185815515325249597961012458, 1.81078263166900856976502893480, 3.16201413700836099058924049943, 3.69112939839868111288315577398, 4.91479620341529252830049884217, 5.37568046673171427264159710817, 5.84873802132001555601248972769, 7.19016113118203639915753456358, 7.78813090126448336529812530859, 8.716681742142359895240910840307

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