Properties

Label 2-2009-1.1-c1-0-57
Degree $2$
Conductor $2009$
Sign $-1$
Analytic cond. $16.0419$
Root an. cond. $4.00523$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.874·2-s − 2.48·3-s − 1.23·4-s − 4.06·5-s − 2.17·6-s − 2.82·8-s + 3.16·9-s − 3.55·10-s + 5.27·11-s + 3.06·12-s + 3.57·13-s + 10.0·15-s − 0.00594·16-s + 4.81·17-s + 2.76·18-s − 5.76·19-s + 5.02·20-s + 4.61·22-s − 6.67·23-s + 7.02·24-s + 11.5·25-s + 3.12·26-s − 0.401·27-s + 2.57·29-s + 8.82·30-s − 7.55·31-s + 5.65·32-s + ⋯
L(s)  = 1  + 0.618·2-s − 1.43·3-s − 0.617·4-s − 1.81·5-s − 0.886·6-s − 1.00·8-s + 1.05·9-s − 1.12·10-s + 1.59·11-s + 0.884·12-s + 0.990·13-s + 2.60·15-s − 0.00148·16-s + 1.16·17-s + 0.651·18-s − 1.32·19-s + 1.12·20-s + 0.983·22-s − 1.39·23-s + 1.43·24-s + 2.30·25-s + 0.612·26-s − 0.0771·27-s + 0.478·29-s + 1.61·30-s − 1.35·31-s + 0.999·32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $-1$
Analytic conductor: \(16.0419\)
Root analytic conductor: \(4.00523\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2009,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.874T + 2T^{2} \)
3 \( 1 + 2.48T + 3T^{2} \)
5 \( 1 + 4.06T + 5T^{2} \)
11 \( 1 - 5.27T + 11T^{2} \)
13 \( 1 - 3.57T + 13T^{2} \)
17 \( 1 - 4.81T + 17T^{2} \)
19 \( 1 + 5.76T + 19T^{2} \)
23 \( 1 + 6.67T + 23T^{2} \)
29 \( 1 - 2.57T + 29T^{2} \)
31 \( 1 + 7.55T + 31T^{2} \)
37 \( 1 - 6.08T + 37T^{2} \)
43 \( 1 + 0.758T + 43T^{2} \)
47 \( 1 - 4.96T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 - 8.86T + 59T^{2} \)
61 \( 1 - 2.58T + 61T^{2} \)
67 \( 1 + 4.34T + 67T^{2} \)
71 \( 1 - 0.0215T + 71T^{2} \)
73 \( 1 - 4.03T + 73T^{2} \)
79 \( 1 + 1.59T + 79T^{2} \)
83 \( 1 + 2.44T + 83T^{2} \)
89 \( 1 + 1.60T + 89T^{2} \)
97 \( 1 + 12.5T + 97T^{2} \)
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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

−8.585669022541447581237971194990, −8.054442843358674163176642828322, −6.91359433334724438418551177179, −6.22229450244393657915245690712, −5.57573085351837107896750749309, −4.42991709934159163684616395953, −4.06345857606614758911116639349, −3.44458035158344830351688838848, −1.08137463809202876393989093276, 0, 1.08137463809202876393989093276, 3.44458035158344830351688838848, 4.06345857606614758911116639349, 4.42991709934159163684616395953, 5.57573085351837107896750749309, 6.22229450244393657915245690712, 6.91359433334724438418551177179, 8.054442843358674163176642828322, 8.585669022541447581237971194990

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