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Properties

Label 2-2005-2005.1202-c0-0-0
Degree $2$
Conductor $2005$
Sign $-0.379 + 0.925i$
Analytic cond. $1.00062$
Root an. cond. $1.00031$
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  + (−1.39 − 1.39i)2-s + 2.90i·4-s + (−0.809 + 0.587i)5-s + (0.642 + 0.642i)7-s + (2.65 − 2.65i)8-s − i·9-s + (1.95 + 0.309i)10-s − 1.90·11-s − 1.79i·14-s − 4.52·16-s + (−1.39 + 1.39i)18-s + (−1.70 − 2.34i)20-s + (2.65 + 2.65i)22-s + (0.309 − 0.951i)25-s + (−1.86 + 1.86i)28-s − 1.61i·29-s + ⋯
L(s)  = 1  + (−1.39 − 1.39i)2-s + 2.90i·4-s + (−0.809 + 0.587i)5-s + (0.642 + 0.642i)7-s + (2.65 − 2.65i)8-s − i·9-s + (1.95 + 0.309i)10-s − 1.90·11-s − 1.79i·14-s − 4.52·16-s + (−1.39 + 1.39i)18-s + (−1.70 − 2.34i)20-s + (2.65 + 2.65i)22-s + (0.309 − 0.951i)25-s + (−1.86 + 1.86i)28-s − 1.61i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2005\)    =    \(5 \cdot 401\)
Sign: $-0.379 + 0.925i$
Analytic conductor: \(1.00062\)
Root analytic conductor: \(1.00031\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2005} (1202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2005,\ (\ :0),\ -0.379 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3788167323\)
\(L(\frac12)\) \(\approx\) \(0.3788167323\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 - 0.587i)T \)
401 \( 1 - T \)
good2 \( 1 + (1.39 + 1.39i)T + iT^{2} \)
3 \( 1 + iT^{2} \)
7 \( 1 + (-0.642 - 0.642i)T + iT^{2} \)
11 \( 1 + 1.90T + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + 1.61iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - 1.17T + T^{2} \)
43 \( 1 + (-0.642 + 0.642i)T - iT^{2} \)
47 \( 1 + (-1.39 - 1.39i)T + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-1.39 + 1.39i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-1 + i)T - iT^{2} \)
89 \( 1 + 1.90iT - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.151650492611581067333268853169, −8.458257352062144417335981163882, −7.71352720439697183232565195475, −7.45956094414223663229399912647, −6.10931382161977412333104688716, −4.63604102969192121960213936168, −3.73771235595019358428299899923, −2.78341634728293976287828378773, −2.25864216647391895235963555492, −0.54471736165815280362200658107, 1.01019034506687226157094589892, 2.33289865574592616128810409188, 4.37339440586349700812311594345, 5.19996054734857370532114565570, 5.46516411890853462956479893588, 6.93830275295892963825396274679, 7.56489002512109170529131658012, 7.952134953960990346372246614656, 8.446384373678406805192925973012, 9.289431451361119830320133659542

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