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 + 0.741·3-s + 4-s + 5-s + 0.741·6-s − 2.92·7-s + 8-s − 2.45·9-s + 10-s + 3.78·11-s + 0.741·12-s − 1.20·13-s − 2.92·14-s + 0.741·15-s + 16-s + 2.24·17-s − 2.45·18-s − 6.75·19-s + 20-s − 2.16·21-s + 3.78·22-s − 2.16·23-s + 0.741·24-s + 25-s − 1.20·26-s − 4.04·27-s − 2.92·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.428·3-s + 0.5·4-s + 0.447·5-s + 0.302·6-s − 1.10·7-s + 0.353·8-s − 0.816·9-s + 0.316·10-s + 1.14·11-s + 0.214·12-s − 0.334·13-s − 0.781·14-s + 0.191·15-s + 0.250·16-s + 0.543·17-s − 0.577·18-s − 1.54·19-s + 0.223·20-s − 0.473·21-s + 0.807·22-s − 0.450·23-s + 0.151·24-s + 0.200·25-s − 0.236·26-s − 0.777·27-s − 0.552·28-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 \( 1 - 0.741T + 3T^{2} \)
7 \( 1 + 2.92T + 7T^{2} \)
11 \( 1 - 3.78T + 11T^{2} \)
13 \( 1 + 1.20T + 13T^{2} \)
17 \( 1 - 2.24T + 17T^{2} \)
19 \( 1 + 6.75T + 19T^{2} \)
23 \( 1 + 2.16T + 23T^{2} \)
29 \( 1 + 7.43T + 29T^{2} \)
31 \( 1 - 2.54T + 31T^{2} \)
37 \( 1 + 2.16T + 37T^{2} \)
41 \( 1 + 4.79T + 41T^{2} \)
43 \( 1 - 2.62T + 43T^{2} \)
47 \( 1 + 9.28T + 47T^{2} \)
53 \( 1 + 4.24T + 53T^{2} \)
59 \( 1 + 2.50T + 59T^{2} \)
61 \( 1 + 2.21T + 61T^{2} \)
67 \( 1 + 6.47T + 67T^{2} \)
71 \( 1 - 3.13T + 71T^{2} \)
73 \( 1 - 7.55T + 73T^{2} \)
79 \( 1 + 7.16T + 79T^{2} \)
83 \( 1 - 14.7T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + 6.45T + 97T^{2} \)
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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

−7.68487734197682835898346618542, −6.66370068154542361600264935047, −6.34113232630156262186877707340, −5.71650058122737448419405725653, −4.80493045775631660510789621290, −3.81350053546426856124603983297, −3.36951505126270498342213169607, −2.48071262002277702703620117053, −1.67607687832385629344424609082, 0, 1.67607687832385629344424609082, 2.48071262002277702703620117053, 3.36951505126270498342213169607, 3.81350053546426856124603983297, 4.80493045775631660510789621290, 5.71650058122737448419405725653, 6.34113232630156262186877707340, 6.66370068154542361600264935047, 7.68487734197682835898346618542

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