Properties

Label 2-2005-1.1-c1-0-43
Degree $2$
Conductor $2005$
Sign $1$
Analytic cond. $16.0100$
Root an. cond. $4.00125$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.77·2-s − 3.16·3-s + 5.67·4-s − 5-s − 8.78·6-s − 2.64·7-s + 10.1·8-s + 7.04·9-s − 2.77·10-s − 1.40·11-s − 17.9·12-s − 1.13·13-s − 7.33·14-s + 3.16·15-s + 16.8·16-s − 2.63·17-s + 19.5·18-s + 7.38·19-s − 5.67·20-s + 8.38·21-s − 3.87·22-s + 4.55·23-s − 32.2·24-s + 25-s − 3.14·26-s − 12.8·27-s − 15.0·28-s + ⋯
L(s)  = 1  + 1.95·2-s − 1.83·3-s + 2.83·4-s − 0.447·5-s − 3.58·6-s − 1.00·7-s + 3.60·8-s + 2.34·9-s − 0.876·10-s − 0.422·11-s − 5.19·12-s − 0.314·13-s − 1.95·14-s + 0.818·15-s + 4.21·16-s − 0.638·17-s + 4.60·18-s + 1.69·19-s − 1.26·20-s + 1.83·21-s − 0.826·22-s + 0.950·23-s − 6.59·24-s + 0.200·25-s − 0.616·26-s − 2.46·27-s − 2.83·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2005\)    =    \(5 \cdot 401\)
Sign: $1$
Analytic conductor: \(16.0100\)
Root analytic conductor: \(4.00125\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2005,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.068087612\)
\(L(\frac12)\) \(\approx\) \(3.068087612\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
401 \( 1 - T \)
good2 \( 1 - 2.77T + 2T^{2} \)
3 \( 1 + 3.16T + 3T^{2} \)
7 \( 1 + 2.64T + 7T^{2} \)
11 \( 1 + 1.40T + 11T^{2} \)
13 \( 1 + 1.13T + 13T^{2} \)
17 \( 1 + 2.63T + 17T^{2} \)
19 \( 1 - 7.38T + 19T^{2} \)
23 \( 1 - 4.55T + 23T^{2} \)
29 \( 1 - 3.85T + 29T^{2} \)
31 \( 1 - 7.86T + 31T^{2} \)
37 \( 1 + 1.77T + 37T^{2} \)
41 \( 1 - 4.80T + 41T^{2} \)
43 \( 1 - 6.91T + 43T^{2} \)
47 \( 1 + 2.52T + 47T^{2} \)
53 \( 1 + 2.06T + 53T^{2} \)
59 \( 1 - 2.37T + 59T^{2} \)
61 \( 1 + 6.66T + 61T^{2} \)
67 \( 1 - 9.83T + 67T^{2} \)
71 \( 1 + 4.63T + 71T^{2} \)
73 \( 1 + 16.9T + 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 - 17.0T + 83T^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 - 9.01T + 97T^{2} \)
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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

−9.599760490246052084573062379430, −7.68529141022403895238304810763, −7.05628617383008388092874900042, −6.43830817855491170188042924192, −5.87588062813756121810958460648, −4.98807282173240749759530662124, −4.65589107766118451397427290465, −3.57803274155435559554297823135, −2.68645676749179331423325386588, −0.979737891810172423484342146491, 0.979737891810172423484342146491, 2.68645676749179331423325386588, 3.57803274155435559554297823135, 4.65589107766118451397427290465, 4.98807282173240749759530662124, 5.87588062813756121810958460648, 6.43830817855491170188042924192, 7.05628617383008388092874900042, 7.68529141022403895238304810763, 9.599760490246052084573062379430

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