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Properties

Label 2-2010-201.200-c1-0-66
Degree $2$
Conductor $2010$
Sign $0.277 + 0.960i$
Analytic cond. $16.0499$
Root an. cond. $4.00623$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (−1.63 + 0.558i)3-s + 4-s + 5-s + (−1.63 + 0.558i)6-s + 2.20i·7-s + 8-s + (2.37 − 1.83i)9-s + 10-s − 6.11·11-s + (−1.63 + 0.558i)12-s − 3.23i·13-s + 2.20i·14-s + (−1.63 + 0.558i)15-s + 16-s − 4.87i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.946 + 0.322i)3-s + 0.5·4-s + 0.447·5-s + (−0.669 + 0.228i)6-s + 0.832i·7-s + 0.353·8-s + (0.791 − 0.610i)9-s + 0.316·10-s − 1.84·11-s + (−0.473 + 0.161i)12-s − 0.897i·13-s + 0.588i·14-s + (−0.423 + 0.144i)15-s + 0.250·16-s − 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2010\)    =    \(2 \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.277 + 0.960i$
Analytic conductor: \(16.0499\)
Root analytic conductor: \(4.00623\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2010} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2010,\ (\ :1/2),\ 0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.395575335\)
\(L(\frac12)\) \(\approx\) \(1.395575335\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.63 - 0.558i)T \)
5 \( 1 - T \)
67 \( 1 + (0.386 - 8.17i)T \)
good7 \( 1 - 2.20iT - 7T^{2} \)
11 \( 1 + 6.11T + 11T^{2} \)
13 \( 1 + 3.23iT - 13T^{2} \)
17 \( 1 + 4.87iT - 17T^{2} \)
19 \( 1 + 1.14T + 19T^{2} \)
23 \( 1 + 5.78iT - 23T^{2} \)
29 \( 1 + 0.744iT - 29T^{2} \)
31 \( 1 + 6.99iT - 31T^{2} \)
37 \( 1 - 1.30T + 37T^{2} \)
41 \( 1 - 7.92T + 41T^{2} \)
43 \( 1 + 1.47iT - 43T^{2} \)
47 \( 1 - 0.615iT - 47T^{2} \)
53 \( 1 + 6.82T + 53T^{2} \)
59 \( 1 - 8.06iT - 59T^{2} \)
61 \( 1 + 3.78iT - 61T^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 - 5.03T + 73T^{2} \)
79 \( 1 + 6.67iT - 79T^{2} \)
83 \( 1 - 2.20iT - 83T^{2} \)
89 \( 1 + 15.6iT - 89T^{2} \)
97 \( 1 - 8.34iT - 97T^{2} \)
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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

−9.157616651391924881936970382296, −8.028380894805085068286138366439, −7.32260829524243106807007089794, −6.21024647490120872696464771735, −5.67436368786037929308432371624, −5.10588084225172605235001751457, −4.40922314313280893335463567102, −2.93470713235905606075232047415, −2.34767759499959319921680973557, −0.43751650316260042810765168702, 1.35141054993190165734566402443, 2.36143217271430295995530908376, 3.69477224712548351692042174318, 4.63378355768485572626196140093, 5.32157116752326452319043497654, 6.02923092639903505591152483759, 6.82686204822840808465675367722, 7.51840536765732338071134321697, 8.224256412906709027563224052920, 9.600252739336396712858503295409

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