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Properties

Label 2-2009-287.40-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  + (0.222 − 0.385i)2-s + (0.900 + 1.56i)3-s + (0.400 + 0.694i)4-s + 0.801·6-s + 0.801·8-s + (−1.12 + 1.94i)9-s + (−0.722 + 1.25i)12-s − 0.445·13-s + (−0.222 + 0.385i)16-s + (−0.623 − 1.07i)17-s + (0.499 + 0.866i)18-s + (−0.623 + 1.07i)19-s + (0.900 − 1.56i)23-s + (0.722 + 1.25i)24-s + (−0.5 − 0.866i)25-s + (−0.0990 + 0.171i)26-s + ⋯
L(s)  = 1  + (0.222 − 0.385i)2-s + (0.900 + 1.56i)3-s + (0.400 + 0.694i)4-s + 0.801·6-s + 0.801·8-s + (−1.12 + 1.94i)9-s + (−0.722 + 1.25i)12-s − 0.445·13-s + (−0.222 + 0.385i)16-s + (−0.623 − 1.07i)17-s + (0.499 + 0.866i)18-s + (−0.623 + 1.07i)19-s + (0.900 − 1.56i)23-s + (0.722 + 1.25i)24-s + (−0.5 − 0.866i)25-s + (−0.0990 + 0.171i)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

\(L(\frac{1}{2})\) \(\approx\) \(1.878283246\)
\(L(\frac12)\) \(\approx\) \(1.878283246\)
\(L(1)\) 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 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + 0.445T + T^{2} \)
17 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 - 1.24T + T^{2} \)
47 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + 1.80T + T^{2} \)
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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

−9.501734500413750520507659063726, −8.870510806783652999768692262194, −8.173531713894780106810018577461, −7.47248128922825924538451269046, −6.39472775744370861980902801105, −5.10424757767847765625776926519, −4.31982938468007056534677439077, −3.89404367349217527185079854756, −2.69947967683693980519869075792, −2.40715702296395911562008266405, 1.22366869102097182474999724637, 2.07155859324925842810246860281, 2.91568693533977308897775536939, 4.20671559624082922823322874354, 5.45861220458254554451597231913, 6.18260824629825547346591162785, 6.94880701234764823031484248432, 7.38689685152472901980093205842, 8.105576416253889765836400360440, 9.063665464194323967344916157578

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