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  + 1.33·2-s − 3-s − 0.214·4-s − 2.78·5-s − 1.33·6-s + 2.26·7-s − 2.95·8-s + 9-s − 3.72·10-s + 2.19·11-s + 0.214·12-s + 4.11·13-s + 3.02·14-s + 2.78·15-s − 3.52·16-s − 17-s + 1.33·18-s − 5.10·19-s + 0.597·20-s − 2.26·21-s + 2.93·22-s − 4.60·23-s + 2.95·24-s + 2.75·25-s + 5.50·26-s − 27-s − 0.486·28-s + ⋯
L(s)  = 1  + 0.944·2-s − 0.577·3-s − 0.107·4-s − 1.24·5-s − 0.545·6-s + 0.857·7-s − 1.04·8-s + 0.333·9-s − 1.17·10-s + 0.662·11-s + 0.0618·12-s + 1.14·13-s + 0.809·14-s + 0.719·15-s − 0.881·16-s − 0.242·17-s + 0.314·18-s − 1.17·19-s + 0.133·20-s − 0.494·21-s + 0.625·22-s − 0.961·23-s + 0.604·24-s + 0.551·25-s + 1.07·26-s − 0.192·27-s − 0.0918·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.597344992$
$L(\frac12)$  $\approx$  $1.597344992$
$L(\frac{3}{2})$   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 \( 1 - 1.33T + 2T^{2} \)
5 \( 1 + 2.78T + 5T^{2} \)
7 \( 1 - 2.26T + 7T^{2} \)
11 \( 1 - 2.19T + 11T^{2} \)
13 \( 1 - 4.11T + 13T^{2} \)
19 \( 1 + 5.10T + 19T^{2} \)
23 \( 1 + 4.60T + 23T^{2} \)
29 \( 1 - 1.32T + 29T^{2} \)
31 \( 1 - 6.45T + 31T^{2} \)
37 \( 1 + 6.29T + 37T^{2} \)
41 \( 1 + 2.12T + 41T^{2} \)
43 \( 1 + 1.24T + 43T^{2} \)
47 \( 1 + 8.03T + 47T^{2} \)
53 \( 1 - 9.17T + 53T^{2} \)
59 \( 1 - 2.78T + 59T^{2} \)
61 \( 1 + 1.20T + 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 - 6.75T + 71T^{2} \)
73 \( 1 - 9.54T + 73T^{2} \)
83 \( 1 - 17.9T + 83T^{2} \)
89 \( 1 - 5.07T + 89T^{2} \)
97 \( 1 - 17.3T + 97T^{2} \)
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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

−8.401141848028310241753000010567, −7.81144062584944063932303256829, −6.53816008332789202719371647885, −6.34550051710836588577201510964, −5.19958151294978912992630786988, −4.62135645238759020516265691571, −3.91083676014107573895762987518, −3.54846174765129568656397378382, −2.01827556884618675201805049791, −0.64790307381119679210632064530, 0.64790307381119679210632064530, 2.01827556884618675201805049791, 3.54846174765129568656397378382, 3.91083676014107573895762987518, 4.62135645238759020516265691571, 5.19958151294978912992630786988, 6.34550051710836588577201510964, 6.53816008332789202719371647885, 7.81144062584944063932303256829, 8.401141848028310241753000010567

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