Properties

Label 2-2005-1.1-c1-0-22
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  + 0.810·2-s − 0.0806·3-s − 1.34·4-s + 5-s − 0.0653·6-s − 2.66·7-s − 2.70·8-s − 2.99·9-s + 0.810·10-s + 0.846·11-s + 0.108·12-s − 2.68·13-s − 2.15·14-s − 0.0806·15-s + 0.490·16-s + 3.80·17-s − 2.42·18-s − 1.65·19-s − 1.34·20-s + 0.214·21-s + 0.686·22-s + 3.67·23-s + 0.218·24-s + 25-s − 2.17·26-s + 0.483·27-s + 3.57·28-s + ⋯
L(s)  = 1  + 0.573·2-s − 0.0465·3-s − 0.671·4-s + 0.447·5-s − 0.0266·6-s − 1.00·7-s − 0.957·8-s − 0.997·9-s + 0.256·10-s + 0.255·11-s + 0.0312·12-s − 0.745·13-s − 0.576·14-s − 0.0208·15-s + 0.122·16-s + 0.922·17-s − 0.571·18-s − 0.379·19-s − 0.300·20-s + 0.0468·21-s + 0.146·22-s + 0.765·23-s + 0.0445·24-s + 0.200·25-s − 0.427·26-s + 0.0929·27-s + 0.676·28-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

\(L(1)\) \(\approx\) \(1.440902935\)
\(L(\frac12)\) \(\approx\) \(1.440902935\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.810T + 2T^{2} \)
3 \( 1 + 0.0806T + 3T^{2} \)
7 \( 1 + 2.66T + 7T^{2} \)
11 \( 1 - 0.846T + 11T^{2} \)
13 \( 1 + 2.68T + 13T^{2} \)
17 \( 1 - 3.80T + 17T^{2} \)
19 \( 1 + 1.65T + 19T^{2} \)
23 \( 1 - 3.67T + 23T^{2} \)
29 \( 1 - 5.56T + 29T^{2} \)
31 \( 1 - 8.93T + 31T^{2} \)
37 \( 1 - 4.44T + 37T^{2} \)
41 \( 1 - 4.31T + 41T^{2} \)
43 \( 1 - 6.96T + 43T^{2} \)
47 \( 1 + 1.19T + 47T^{2} \)
53 \( 1 - 0.743T + 53T^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 + 3.87T + 61T^{2} \)
67 \( 1 - 6.88T + 67T^{2} \)
71 \( 1 + 3.28T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + 0.745T + 79T^{2} \)
83 \( 1 - 0.345T + 83T^{2} \)
89 \( 1 - 3.91T + 89T^{2} \)
97 \( 1 - 5.92T + 97T^{2} \)
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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

−9.218412430691534803986654426430, −8.544547516206221374713127322690, −7.60650771308422007994387740151, −6.36876852435798913476707241170, −6.03857053534242528105769549593, −5.10397221815294148968494706283, −4.35585996545991847959126601807, −3.14737162376881430778902712762, −2.72376340677201062428327803079, −0.72199884147993809593394097408, 0.72199884147993809593394097408, 2.72376340677201062428327803079, 3.14737162376881430778902712762, 4.35585996545991847959126601807, 5.10397221815294148968494706283, 6.03857053534242528105769549593, 6.36876852435798913476707241170, 7.60650771308422007994387740151, 8.544547516206221374713127322690, 9.218412430691534803986654426430

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