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 + 0.844·3-s + 4-s + 5-s + 0.844·6-s − 0.532·7-s + 8-s − 2.28·9-s + 10-s − 0.961·11-s + 0.844·12-s − 1.20·13-s − 0.532·14-s + 0.844·15-s + 16-s − 5.49·17-s − 2.28·18-s − 4.34·19-s + 20-s − 0.449·21-s − 0.961·22-s + 6.60·23-s + 0.844·24-s + 25-s − 1.20·26-s − 4.46·27-s − 0.532·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.487·3-s + 0.5·4-s + 0.447·5-s + 0.344·6-s − 0.201·7-s + 0.353·8-s − 0.762·9-s + 0.316·10-s − 0.289·11-s + 0.243·12-s − 0.333·13-s − 0.142·14-s + 0.217·15-s + 0.250·16-s − 1.33·17-s − 0.539·18-s − 0.995·19-s + 0.223·20-s − 0.0981·21-s − 0.204·22-s + 1.37·23-s + 0.172·24-s + 0.200·25-s − 0.235·26-s − 0.858·27-s − 0.100·28-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 \( 1 - 0.844T + 3T^{2} \)
7 \( 1 + 0.532T + 7T^{2} \)
11 \( 1 + 0.961T + 11T^{2} \)
13 \( 1 + 1.20T + 13T^{2} \)
17 \( 1 + 5.49T + 17T^{2} \)
19 \( 1 + 4.34T + 19T^{2} \)
23 \( 1 - 6.60T + 23T^{2} \)
29 \( 1 + 7.33T + 29T^{2} \)
31 \( 1 - 7.02T + 31T^{2} \)
37 \( 1 + 5.84T + 37T^{2} \)
41 \( 1 - 9.20T + 41T^{2} \)
43 \( 1 + 9.44T + 43T^{2} \)
47 \( 1 + 7.28T + 47T^{2} \)
53 \( 1 - 6.24T + 53T^{2} \)
59 \( 1 + 9.77T + 59T^{2} \)
61 \( 1 + 5.61T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + 3.84T + 71T^{2} \)
73 \( 1 - 7.54T + 73T^{2} \)
79 \( 1 + 0.0908T + 79T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 - 8.35T + 97T^{2} \)
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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

−7.71066385931601114359963352606, −6.75945909291078337881280076431, −6.39565829533243571735646491794, −5.46837836190352683170926718212, −4.87038563315760040424114204989, −4.05462561895886368744117856837, −3.08744009158979899294256990025, −2.54005544020896545523220619531, −1.72724761137716556074125977196, 0, 1.72724761137716556074125977196, 2.54005544020896545523220619531, 3.08744009158979899294256990025, 4.05462561895886368744117856837, 4.87038563315760040424114204989, 5.46837836190352683170926718212, 6.39565829533243571735646491794, 6.75945909291078337881280076431, 7.71066385931601114359963352606

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