Properties

Degree 2
Conductor $ 7^{2} \cdot 41 $
Sign $0.0633 - 0.997i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.623 − 1.07i)2-s + (−0.222 + 0.385i)3-s + (−0.277 + 0.480i)4-s + 0.554·6-s − 0.554·8-s + (0.400 + 0.694i)9-s + (−0.123 − 0.213i)12-s − 1.24·13-s + (0.623 + 1.07i)16-s + (−0.900 + 1.56i)17-s + (0.5 − 0.866i)18-s + (−0.900 − 1.56i)19-s + (0.222 + 0.385i)23-s + (0.123 − 0.213i)24-s + (−0.5 + 0.866i)25-s + (0.777 + 1.34i)26-s + ⋯
L(s)  = 1  + (−0.623 − 1.07i)2-s + (−0.222 + 0.385i)3-s + (−0.277 + 0.480i)4-s + 0.554·6-s − 0.554·8-s + (0.400 + 0.694i)9-s + (−0.123 − 0.213i)12-s − 1.24·13-s + (0.623 + 1.07i)16-s + (−0.900 + 1.56i)17-s + (0.5 − 0.866i)18-s + (−0.900 − 1.56i)19-s + (0.222 + 0.385i)23-s + (0.123 − 0.213i)24-s + (−0.5 + 0.866i)25-s + (0.777 + 1.34i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2009\)    =    \(7^{2} \cdot 41\)
\( \varepsilon \)  =  $0.0633 - 0.997i$
motivic weight  =  \(0\)
character  :  $\chi_{2009} (1844, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2009,\ (\ :0),\ 0.0633 - 0.997i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.2403303170\)
\(L(\frac12)\)  \(\approx\)  \(0.2403303170\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{7,\;41\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 + T \)
good2 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + 1.24T + T^{2} \)
17 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + 1.80T + T^{2} \)
47 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - 0.445T + T^{2} \)
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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

−9.643561966556845884824569782185, −9.027426783287319436826280660516, −8.275287345957190598706870319513, −7.27167007435329756697061110066, −6.44225908984818691739280741230, −5.34409398241446095882177459537, −4.55516777346898850179205652688, −3.60603968135986095328287220566, −2.38895967264862872838212690630, −1.76580778284001710353102003885, 0.20020009897710312996831977629, 2.03449118281866938440195291613, 3.24868771091844489836162223007, 4.50092349371456262817860635579, 5.39069365818424957350769252050, 6.42305358294661860649159643141, 6.80498625389605935836623737817, 7.46481482377497099278194546863, 8.298547009856418060094801826824, 8.916800549862688359453662619855

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