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  + 2.13·2-s − 3-s + 2.57·4-s − 3.64·5-s − 2.13·6-s − 4.64·7-s + 1.22·8-s + 9-s − 7.78·10-s − 0.795·11-s − 2.57·12-s − 0.469·13-s − 9.92·14-s + 3.64·15-s − 2.52·16-s − 17-s + 2.13·18-s + 3.75·19-s − 9.36·20-s + 4.64·21-s − 1.70·22-s − 5.28·23-s − 1.22·24-s + 8.25·25-s − 1.00·26-s − 27-s − 11.9·28-s + ⋯
L(s)  = 1  + 1.51·2-s − 0.577·3-s + 1.28·4-s − 1.62·5-s − 0.872·6-s − 1.75·7-s + 0.432·8-s + 0.333·9-s − 2.46·10-s − 0.239·11-s − 0.742·12-s − 0.130·13-s − 2.65·14-s + 0.940·15-s − 0.632·16-s − 0.242·17-s + 0.503·18-s + 0.860·19-s − 2.09·20-s + 1.01·21-s − 0.362·22-s − 1.10·23-s − 0.249·24-s + 1.65·25-s − 0.196·26-s − 0.192·27-s − 2.25·28-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  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.123963077$
$L(\frac12)$  $\approx$  $1.123963077$
$L(\frac{3}{2})$   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 \( 1 - 2.13T + 2T^{2} \)
5 \( 1 + 3.64T + 5T^{2} \)
7 \( 1 + 4.64T + 7T^{2} \)
11 \( 1 + 0.795T + 11T^{2} \)
13 \( 1 + 0.469T + 13T^{2} \)
19 \( 1 - 3.75T + 19T^{2} \)
23 \( 1 + 5.28T + 23T^{2} \)
29 \( 1 - 1.22T + 29T^{2} \)
31 \( 1 - 3.94T + 31T^{2} \)
37 \( 1 - 3.13T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 - 3.77T + 43T^{2} \)
47 \( 1 - 1.22T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + 0.135T + 59T^{2} \)
61 \( 1 - 7.09T + 61T^{2} \)
67 \( 1 + 0.755T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 - 9.13T + 73T^{2} \)
83 \( 1 - 4.90T + 83T^{2} \)
89 \( 1 - 17.8T + 89T^{2} \)
97 \( 1 - 6.14T + 97T^{2} \)
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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

−8.164450928465752843647631252501, −7.42605944151744871889081944054, −6.61609338943054481439798970976, −6.30057701116770547123552490962, −5.30654122043749210076405537712, −4.61154384623744366702021319633, −3.73694685144579931650159738769, −3.45792130130296208399254999172, −2.54580783717044748219705007174, −0.46499392726948219064697004202, 0.46499392726948219064697004202, 2.54580783717044748219705007174, 3.45792130130296208399254999172, 3.73694685144579931650159738769, 4.61154384623744366702021319633, 5.30654122043749210076405537712, 6.30057701116770547123552490962, 6.61609338943054481439798970976, 7.42605944151744871889081944054, 8.164450928465752843647631252501

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