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 − 1.13·3-s + 4-s + 5-s − 1.13·6-s − 1.31·7-s + 8-s − 1.70·9-s + 10-s − 0.369·11-s − 1.13·12-s − 5.82·13-s − 1.31·14-s − 1.13·15-s + 16-s + 6.05·17-s − 1.70·18-s + 0.866·19-s + 20-s + 1.49·21-s − 0.369·22-s − 1.64·23-s − 1.13·24-s + 25-s − 5.82·26-s + 5.35·27-s − 1.31·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.655·3-s + 0.5·4-s + 0.447·5-s − 0.463·6-s − 0.497·7-s + 0.353·8-s − 0.569·9-s + 0.316·10-s − 0.111·11-s − 0.327·12-s − 1.61·13-s − 0.351·14-s − 0.293·15-s + 0.250·16-s + 1.46·17-s − 0.402·18-s + 0.198·19-s + 0.223·20-s + 0.326·21-s − 0.0788·22-s − 0.343·23-s − 0.231·24-s + 0.200·25-s − 1.14·26-s + 1.02·27-s − 0.248·28-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 \( 1 + 1.13T + 3T^{2} \)
7 \( 1 + 1.31T + 7T^{2} \)
11 \( 1 + 0.369T + 11T^{2} \)
13 \( 1 + 5.82T + 13T^{2} \)
17 \( 1 - 6.05T + 17T^{2} \)
19 \( 1 - 0.866T + 19T^{2} \)
23 \( 1 + 1.64T + 23T^{2} \)
29 \( 1 - 7.34T + 29T^{2} \)
31 \( 1 + 1.21T + 31T^{2} \)
37 \( 1 + 2.26T + 37T^{2} \)
41 \( 1 - 0.302T + 41T^{2} \)
43 \( 1 - 9.60T + 43T^{2} \)
47 \( 1 + 0.473T + 47T^{2} \)
53 \( 1 + 5.12T + 53T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 + 14.7T + 61T^{2} \)
67 \( 1 + 15.8T + 67T^{2} \)
71 \( 1 + 16.2T + 71T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + 14.7T + 83T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 + 13.0T + 97T^{2} \)
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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

−7.51087993806648948430459262201, −6.87868910448749272698564286014, −6.06710181249486061634453998933, −5.58466059065779839481459550956, −5.00160066510393926517600295637, −4.23586685082954097964969207389, −2.97503542510291138184186482898, −2.72703231516226233184829136066, −1.35579958479508762978357782730, 0, 1.35579958479508762978357782730, 2.72703231516226233184829136066, 2.97503542510291138184186482898, 4.23586685082954097964969207389, 5.00160066510393926517600295637, 5.58466059065779839481459550956, 6.06710181249486061634453998933, 6.87868910448749272698564286014, 7.51087993806648948430459262201

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