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Properties

Label 2-2009-287.40-c0-0-4
Degree $2$
Conductor $2009$
Sign $0.0633 + 0.997i$
Analytic cond. $1.00262$
Root an. cond. $1.00130$
Motivic weight $0$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $0.0633 + 0.997i$
Analytic conductor: \(1.00262\)
Root analytic conductor: \(1.00130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2009} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2009,\ (\ :0),\ 0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2403303170\)
\(L(\frac12)\) \(\approx\) \(0.2403303170\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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