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 − 0.369·3-s + 4-s + 5-s − 0.369·6-s + 4.16·7-s + 8-s − 2.86·9-s + 10-s − 5.82·11-s − 0.369·12-s − 5.41·13-s + 4.16·14-s − 0.369·15-s + 16-s − 3.34·17-s − 2.86·18-s + 1.75·19-s + 20-s − 1.53·21-s − 5.82·22-s + 1.42·23-s − 0.369·24-s + 25-s − 5.41·26-s + 2.16·27-s + 4.16·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.213·3-s + 0.5·4-s + 0.447·5-s − 0.150·6-s + 1.57·7-s + 0.353·8-s − 0.954·9-s + 0.316·10-s − 1.75·11-s − 0.106·12-s − 1.50·13-s + 1.11·14-s − 0.0953·15-s + 0.250·16-s − 0.811·17-s − 0.674·18-s + 0.402·19-s + 0.223·20-s − 0.335·21-s − 1.24·22-s + 0.297·23-s − 0.0753·24-s + 0.200·25-s − 1.06·26-s + 0.416·27-s + 0.786·28-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 \( 1 + 0.369T + 3T^{2} \)
7 \( 1 - 4.16T + 7T^{2} \)
11 \( 1 + 5.82T + 11T^{2} \)
13 \( 1 + 5.41T + 13T^{2} \)
17 \( 1 + 3.34T + 17T^{2} \)
19 \( 1 - 1.75T + 19T^{2} \)
23 \( 1 - 1.42T + 23T^{2} \)
29 \( 1 - 3.08T + 29T^{2} \)
31 \( 1 - 0.619T + 31T^{2} \)
37 \( 1 - 7.58T + 37T^{2} \)
41 \( 1 + 4.82T + 41T^{2} \)
43 \( 1 + 0.960T + 43T^{2} \)
47 \( 1 + 6.56T + 47T^{2} \)
53 \( 1 + 2.02T + 53T^{2} \)
59 \( 1 + 11.0T + 59T^{2} \)
61 \( 1 + 9.76T + 61T^{2} \)
67 \( 1 - 7.72T + 67T^{2} \)
71 \( 1 + 7.24T + 71T^{2} \)
73 \( 1 + 7.65T + 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + 9.11T + 83T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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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

−7.80110179394539371055740142702, −7.03299424885819491149305958850, −6.05599104463914251060979121363, −5.32792793850577420304013572448, −4.91607300012698081941212197560, −4.49981748915334164583039269030, −2.85424768843153241687971640864, −2.59780507756580884590120047144, −1.61391756827453644507911255741, 0, 1.61391756827453644507911255741, 2.59780507756580884590120047144, 2.85424768843153241687971640864, 4.49981748915334164583039269030, 4.91607300012698081941212197560, 5.32792793850577420304013572448, 6.05599104463914251060979121363, 7.03299424885819491149305958850, 7.80110179394539371055740142702

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