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 − 2.23·3-s + 4-s + 5-s − 2.23·6-s − 0.971·7-s + 8-s + 1.99·9-s + 10-s − 2.26·11-s − 2.23·12-s + 4.74·13-s − 0.971·14-s − 2.23·15-s + 16-s − 5.16·17-s + 1.99·18-s − 2.10·19-s + 20-s + 2.17·21-s − 2.26·22-s + 3.32·23-s − 2.23·24-s + 25-s + 4.74·26-s + 2.24·27-s − 0.971·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.29·3-s + 0.5·4-s + 0.447·5-s − 0.912·6-s − 0.367·7-s + 0.353·8-s + 0.664·9-s + 0.316·10-s − 0.684·11-s − 0.645·12-s + 1.31·13-s − 0.259·14-s − 0.577·15-s + 0.250·16-s − 1.25·17-s + 0.470·18-s − 0.482·19-s + 0.223·20-s + 0.474·21-s − 0.483·22-s + 0.693·23-s − 0.456·24-s + 0.200·25-s + 0.931·26-s + 0.432·27-s − 0.183·28-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 \( 1 + 2.23T + 3T^{2} \)
7 \( 1 + 0.971T + 7T^{2} \)
11 \( 1 + 2.26T + 11T^{2} \)
13 \( 1 - 4.74T + 13T^{2} \)
17 \( 1 + 5.16T + 17T^{2} \)
19 \( 1 + 2.10T + 19T^{2} \)
23 \( 1 - 3.32T + 23T^{2} \)
29 \( 1 + 7.06T + 29T^{2} \)
31 \( 1 - 5.27T + 31T^{2} \)
37 \( 1 + 6.88T + 37T^{2} \)
41 \( 1 - 11.4T + 41T^{2} \)
43 \( 1 - 5.15T + 43T^{2} \)
47 \( 1 + 2.01T + 47T^{2} \)
53 \( 1 + 9.34T + 53T^{2} \)
59 \( 1 - 8.51T + 59T^{2} \)
61 \( 1 + 3.05T + 61T^{2} \)
67 \( 1 + 4.20T + 67T^{2} \)
71 \( 1 + 3.81T + 71T^{2} \)
73 \( 1 + 13.6T + 73T^{2} \)
79 \( 1 + 14.8T + 79T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 - 2.24T + 89T^{2} \)
97 \( 1 + 8.88T + 97T^{2} \)
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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

−7.39184696072079367377925165119, −6.69665716263825871679888343351, −6.03661394810270604147196214193, −5.80063138172291776044703895429, −4.87867383424718891826193881659, −4.32316240086379864259877887290, −3.31000262137355842980454493013, −2.39113271055716841462225442808, −1.29340191557793768935033068759, 0, 1.29340191557793768935033068759, 2.39113271055716841462225442808, 3.31000262137355842980454493013, 4.32316240086379864259877887290, 4.87867383424718891826193881659, 5.80063138172291776044703895429, 6.03661394810270604147196214193, 6.69665716263825871679888343351, 7.39184696072079367377925165119

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