Properties

Label 2-2009-287.122-c0-0-1
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  + (0.900 + 1.56i)2-s + (−0.623 + 1.07i)3-s + (−1.12 + 1.94i)4-s − 2.24·6-s − 2.24·8-s + (−0.277 − 0.480i)9-s + (−1.40 − 2.42i)12-s − 1.80·13-s + (−0.900 − 1.56i)16-s + (0.222 − 0.385i)17-s + (0.5 − 0.866i)18-s + (0.222 + 0.385i)19-s + (−0.623 − 1.07i)23-s + (1.40 − 2.42i)24-s + (−0.5 + 0.866i)25-s + (−1.62 − 2.81i)26-s + ⋯
L(s)  = 1  + (0.900 + 1.56i)2-s + (−0.623 + 1.07i)3-s + (−1.12 + 1.94i)4-s − 2.24·6-s − 2.24·8-s + (−0.277 − 0.480i)9-s + (−1.40 − 2.42i)12-s − 1.80·13-s + (−0.900 − 1.56i)16-s + (0.222 − 0.385i)17-s + (0.5 − 0.866i)18-s + (0.222 + 0.385i)19-s + (−0.623 − 1.07i)23-s + (1.40 − 2.42i)24-s + (−0.5 + 0.866i)25-s + (−1.62 − 2.81i)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: $\chi_{2009} (1844, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2009,\ (\ :0),\ -0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.002679729\)
\(L(\frac12)\) \(\approx\) \(1.002679729\)
\(L(1)\) 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 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + 1.80T + T^{2} \)
17 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + 0.445T + T^{2} \)
47 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - 1.24T + T^{2} \)
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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

−9.745594627877668701278594924418, −9.158477423166957098957490893041, −7.903282802228966395340160906042, −7.56431462177653868759066503044, −6.55812162235693510804285214602, −5.81953676659684135753179232953, −5.06076217871056130586479215509, −4.62354146552116536650053120354, −3.88736094479582720847191612811, −2.71033009321280303841332942819, 0.55953660300118954990355198473, 1.89413076196815578054865504404, 2.46986856091967004084412632515, 3.69485738022074715154526430086, 4.56913344856577448702948722213, 5.48795262029539389189878875490, 6.01661406167236716670659082192, 7.17652178202595137530379775050, 7.75094982390089087475836397043, 9.164365963390190177880734472087

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