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  + 2.74·2-s − 3-s + 5.55·4-s + 0.373·5-s − 2.74·6-s − 2.56·7-s + 9.76·8-s + 9-s + 1.02·10-s + 5.33·11-s − 5.55·12-s + 1.62·13-s − 7.05·14-s − 0.373·15-s + 15.7·16-s − 17-s + 2.74·18-s + 6.40·19-s + 2.07·20-s + 2.56·21-s + 14.6·22-s − 8.29·23-s − 9.76·24-s − 4.86·25-s + 4.47·26-s − 27-s − 14.2·28-s + ⋯
L(s)  = 1  + 1.94·2-s − 0.577·3-s + 2.77·4-s + 0.167·5-s − 1.12·6-s − 0.970·7-s + 3.45·8-s + 0.333·9-s + 0.324·10-s + 1.60·11-s − 1.60·12-s + 0.451·13-s − 1.88·14-s − 0.0965·15-s + 3.93·16-s − 0.242·17-s + 0.647·18-s + 1.46·19-s + 0.464·20-s + 0.560·21-s + 3.12·22-s − 1.72·23-s − 1.99·24-s − 0.972·25-s + 0.877·26-s − 0.192·27-s − 2.69·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.399388287$
$L(\frac12)$  $\approx$  $6.399388287$
$L(\frac{3}{2})$   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 \( 1 - 2.74T + 2T^{2} \)
5 \( 1 - 0.373T + 5T^{2} \)
7 \( 1 + 2.56T + 7T^{2} \)
11 \( 1 - 5.33T + 11T^{2} \)
13 \( 1 - 1.62T + 13T^{2} \)
19 \( 1 - 6.40T + 19T^{2} \)
23 \( 1 + 8.29T + 23T^{2} \)
29 \( 1 - 7.11T + 29T^{2} \)
31 \( 1 + 0.319T + 31T^{2} \)
37 \( 1 + 9.68T + 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 + 2.22T + 43T^{2} \)
47 \( 1 + 1.32T + 47T^{2} \)
53 \( 1 - 0.149T + 53T^{2} \)
59 \( 1 - 0.248T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 6.82T + 67T^{2} \)
71 \( 1 + 9.62T + 71T^{2} \)
73 \( 1 - 1.57T + 73T^{2} \)
83 \( 1 - 17.5T + 83T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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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

−8.141802131127538057807583432408, −7.14966755031088035396126077165, −6.59189244593073322585987356215, −6.09225486916085199874215698158, −5.57611466510523236900675656044, −4.61654251526713957398100495353, −3.78979192157965765833063298313, −3.47105572823660810944865511207, −2.25957639483167927629321039514, −1.23121136207823174344314831860, 1.23121136207823174344314831860, 2.25957639483167927629321039514, 3.47105572823660810944865511207, 3.78979192157965765833063298313, 4.61654251526713957398100495353, 5.57611466510523236900675656044, 6.09225486916085199874215698158, 6.59189244593073322585987356215, 7.14966755031088035396126077165, 8.141802131127538057807583432408

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