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 + 0.527·3-s + 4-s + 5-s + 0.527·6-s − 1.45·7-s + 8-s − 2.72·9-s + 10-s − 2.62·11-s + 0.527·12-s + 1.24·13-s − 1.45·14-s + 0.527·15-s + 16-s + 2.45·17-s − 2.72·18-s − 1.20·19-s + 20-s − 0.768·21-s − 2.62·22-s − 1.35·23-s + 0.527·24-s + 25-s + 1.24·26-s − 3.01·27-s − 1.45·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.304·3-s + 0.5·4-s + 0.447·5-s + 0.215·6-s − 0.550·7-s + 0.353·8-s − 0.907·9-s + 0.316·10-s − 0.790·11-s + 0.152·12-s + 0.344·13-s − 0.389·14-s + 0.136·15-s + 0.250·16-s + 0.596·17-s − 0.641·18-s − 0.276·19-s + 0.223·20-s − 0.167·21-s − 0.558·22-s − 0.282·23-s + 0.107·24-s + 0.200·25-s + 0.243·26-s − 0.581·27-s − 0.275·28-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 \( 1 - 0.527T + 3T^{2} \)
7 \( 1 + 1.45T + 7T^{2} \)
11 \( 1 + 2.62T + 11T^{2} \)
13 \( 1 - 1.24T + 13T^{2} \)
17 \( 1 - 2.45T + 17T^{2} \)
19 \( 1 + 1.20T + 19T^{2} \)
23 \( 1 + 1.35T + 23T^{2} \)
29 \( 1 - 0.521T + 29T^{2} \)
31 \( 1 + 2.53T + 31T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + 4.79T + 41T^{2} \)
43 \( 1 + 6.82T + 43T^{2} \)
47 \( 1 - 7.76T + 47T^{2} \)
53 \( 1 - 2.85T + 53T^{2} \)
59 \( 1 - 4.64T + 59T^{2} \)
61 \( 1 + 5.90T + 61T^{2} \)
67 \( 1 + 2.08T + 67T^{2} \)
71 \( 1 - 9.96T + 71T^{2} \)
73 \( 1 + 5.44T + 73T^{2} \)
79 \( 1 + 9.71T + 79T^{2} \)
83 \( 1 + 17.4T + 83T^{2} \)
89 \( 1 - 5.36T + 89T^{2} \)
97 \( 1 + 13.4T + 97T^{2} \)
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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

−7.66638311383741721753648462013, −6.89187263305276328057551086786, −6.16622583253131947123235023377, −5.51743519678368758750229640697, −5.04236885821256279367241968855, −3.86599129981886267847614435739, −3.21847513165867978101153912966, −2.56615207187570350302756839160, −1.62991744800477580005902871612, 0, 1.62991744800477580005902871612, 2.56615207187570350302756839160, 3.21847513165867978101153912966, 3.86599129981886267847614435739, 5.04236885821256279367241968855, 5.51743519678368758750229640697, 6.16622583253131947123235023377, 6.89187263305276328057551086786, 7.66638311383741721753648462013

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