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 + 1.99·7-s + 8-s + 2.20·9-s + 10-s + 1.87·11-s − 2.28·12-s − 4.38·13-s + 1.99·14-s − 2.28·15-s + 16-s − 2.44·17-s + 2.20·18-s − 5.14·19-s + 20-s − 4.54·21-s + 1.87·22-s + 5.32·23-s − 2.28·24-s + 25-s − 4.38·26-s + 1.81·27-s + 1.99·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.31·3-s + 0.5·4-s + 0.447·5-s − 0.931·6-s + 0.753·7-s + 0.353·8-s + 0.734·9-s + 0.316·10-s + 0.565·11-s − 0.658·12-s − 1.21·13-s + 0.533·14-s − 0.589·15-s + 0.250·16-s − 0.593·17-s + 0.519·18-s − 1.17·19-s + 0.223·20-s − 0.992·21-s + 0.399·22-s + 1.11·23-s − 0.465·24-s + 0.200·25-s − 0.859·26-s + 0.349·27-s + 0.376·28-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 \( 1 - 1.99T + 7T^{2} \)
11 \( 1 - 1.87T + 11T^{2} \)
13 \( 1 + 4.38T + 13T^{2} \)
17 \( 1 + 2.44T + 17T^{2} \)
19 \( 1 + 5.14T + 19T^{2} \)
23 \( 1 - 5.32T + 23T^{2} \)
29 \( 1 - 8.18T + 29T^{2} \)
31 \( 1 + 9.68T + 31T^{2} \)
37 \( 1 + 4.69T + 37T^{2} \)
41 \( 1 + 2.34T + 41T^{2} \)
43 \( 1 + 9.31T + 43T^{2} \)
47 \( 1 + 1.00T + 47T^{2} \)
53 \( 1 + 3.26T + 53T^{2} \)
59 \( 1 + 2.78T + 59T^{2} \)
61 \( 1 + 0.527T + 61T^{2} \)
67 \( 1 - 6.25T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + 9.22T + 73T^{2} \)
79 \( 1 + 4.06T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 - 1.69T + 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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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

−7.34254677629918493851146777431, −6.75776859713302252636155479272, −6.28232727093069300525785980743, −5.43773937935169869501722424899, −4.79401854728738404571150915609, −4.58388998368033834367874908696, −3.31301092384173760206926827729, −2.23031106235009458147512756455, −1.41828144640829974511381546583, 0, 1.41828144640829974511381546583, 2.23031106235009458147512756455, 3.31301092384173760206926827729, 4.58388998368033834367874908696, 4.79401854728738404571150915609, 5.43773937935169869501722424899, 6.28232727093069300525785980743, 6.75776859713302252636155479272, 7.34254677629918493851146777431

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