Properties

Label 2-2005-1.1-c1-0-96
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.13·2-s + 1.65·3-s + 2.56·4-s − 5-s − 3.53·6-s + 1.78·7-s − 1.19·8-s − 0.257·9-s + 2.13·10-s + 1.00·11-s + 4.24·12-s + 1.91·13-s − 3.80·14-s − 1.65·15-s − 2.56·16-s − 4.60·17-s + 0.549·18-s + 0.868·19-s − 2.56·20-s + 2.95·21-s − 2.13·22-s − 9.35·23-s − 1.98·24-s + 25-s − 4.08·26-s − 5.39·27-s + 4.56·28-s + ⋯
L(s)  = 1  − 1.51·2-s + 0.956·3-s + 1.28·4-s − 0.447·5-s − 1.44·6-s + 0.673·7-s − 0.423·8-s − 0.0858·9-s + 0.675·10-s + 0.301·11-s + 1.22·12-s + 0.530·13-s − 1.01·14-s − 0.427·15-s − 0.641·16-s − 1.11·17-s + 0.129·18-s + 0.199·19-s − 0.572·20-s + 0.644·21-s − 0.455·22-s − 1.95·23-s − 0.404·24-s + 0.200·25-s − 0.801·26-s − 1.03·27-s + 0.862·28-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: \(1\)
Selberg data: \((2,\ 2005,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 + T \)
401 \( 1 + T \)
good2 \( 1 + 2.13T + 2T^{2} \)
3 \( 1 - 1.65T + 3T^{2} \)
7 \( 1 - 1.78T + 7T^{2} \)
11 \( 1 - 1.00T + 11T^{2} \)
13 \( 1 - 1.91T + 13T^{2} \)
17 \( 1 + 4.60T + 17T^{2} \)
19 \( 1 - 0.868T + 19T^{2} \)
23 \( 1 + 9.35T + 23T^{2} \)
29 \( 1 - 0.573T + 29T^{2} \)
31 \( 1 + 6.48T + 31T^{2} \)
37 \( 1 + 4.65T + 37T^{2} \)
41 \( 1 + 4.22T + 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 + 6.38T + 47T^{2} \)
53 \( 1 - 6.43T + 53T^{2} \)
59 \( 1 + 5.73T + 59T^{2} \)
61 \( 1 - 8.80T + 61T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 - 7.25T + 71T^{2} \)
73 \( 1 - 7.85T + 73T^{2} \)
79 \( 1 + 7.28T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 + 3.96T + 97T^{2} \)
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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

−8.644098232221662120009429559421, −8.257426336952738113390917782048, −7.62380071201340241171236423547, −6.85823610484573133217891913538, −5.79752158232357882868617204249, −4.44237067034606699966324141393, −3.62253800451073475171938358272, −2.33014912370423811939972042347, −1.60732586358733182596639695628, 0, 1.60732586358733182596639695628, 2.33014912370423811939972042347, 3.62253800451073475171938358272, 4.44237067034606699966324141393, 5.79752158232357882868617204249, 6.85823610484573133217891913538, 7.62380071201340241171236423547, 8.257426336952738113390917782048, 8.644098232221662120009429559421

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