Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 601 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2.28·3-s + 4-s + 5-s + 2.28·6-s − 3.46·7-s + 8-s + 2.22·9-s + 10-s − 3.41·11-s + 2.28·12-s − 2.63·13-s − 3.46·14-s + 2.28·15-s + 16-s − 3.45·17-s + 2.22·18-s + 1.07·19-s + 20-s − 7.93·21-s − 3.41·22-s − 6.75·23-s + 2.28·24-s + 25-s − 2.63·26-s − 1.76·27-s − 3.46·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.32·3-s + 0.5·4-s + 0.447·5-s + 0.933·6-s − 1.31·7-s + 0.353·8-s + 0.743·9-s + 0.316·10-s − 1.03·11-s + 0.660·12-s − 0.731·13-s − 0.926·14-s + 0.590·15-s + 0.250·16-s − 0.839·17-s + 0.525·18-s + 0.246·19-s + 0.223·20-s − 1.73·21-s − 0.728·22-s − 1.40·23-s + 0.466·24-s + 0.200·25-s − 0.517·26-s − 0.338·27-s − 0.655·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6010\)    =    \(2 \cdot 5 \cdot 601\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6010} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6010,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;601\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;601\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
5 \( 1 - T \)
601 \( 1 + T \)
good3 \( 1 - 2.28T + 3T^{2} \)
7 \( 1 + 3.46T + 7T^{2} \)
11 \( 1 + 3.41T + 11T^{2} \)
13 \( 1 + 2.63T + 13T^{2} \)
17 \( 1 + 3.45T + 17T^{2} \)
19 \( 1 - 1.07T + 19T^{2} \)
23 \( 1 + 6.75T + 23T^{2} \)
29 \( 1 + 1.95T + 29T^{2} \)
31 \( 1 - 4.65T + 31T^{2} \)
37 \( 1 + 6.85T + 37T^{2} \)
41 \( 1 - 0.892T + 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 - 2.60T + 47T^{2} \)
53 \( 1 + 14.2T + 53T^{2} \)
59 \( 1 + 9.44T + 59T^{2} \)
61 \( 1 - 9.42T + 61T^{2} \)
67 \( 1 - 9.09T + 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 - 3.96T + 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 0.456T + 89T^{2} \)
97 \( 1 - 16.6T + 97T^{2} \)
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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

−7.67062652150338943877844611368, −7.07374060030494468590581495929, −6.23526711869542201110190693254, −5.65525317435326624189783764453, −4.67933015326545087802850279747, −3.88757901100871897122039301836, −3.05825527955010011722365052422, −2.60054270983711483376143536961, −1.92201665217933396262740860895, 0, 1.92201665217933396262740860895, 2.60054270983711483376143536961, 3.05825527955010011722365052422, 3.88757901100871897122039301836, 4.67933015326545087802850279747, 5.65525317435326624189783764453, 6.23526711869542201110190693254, 7.07374060030494468590581495929, 7.67062652150338943877844611368

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