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Properties

Label 2-2010-201.200-c1-0-71
Degree $2$
Conductor $2010$
Sign $0.984 + 0.173i$
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 + (1.48 + 0.895i)3-s + 4-s + 5-s + (1.48 + 0.895i)6-s − 3.66i·7-s + 8-s + (1.39 + 2.65i)9-s + 10-s − 1.69·11-s + (1.48 + 0.895i)12-s − 4.67i·13-s − 3.66i·14-s + (1.48 + 0.895i)15-s + 16-s + 1.90i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.855 + 0.517i)3-s + 0.5·4-s + 0.447·5-s + (0.605 + 0.365i)6-s − 1.38i·7-s + 0.353·8-s + (0.464 + 0.885i)9-s + 0.316·10-s − 0.509·11-s + (0.427 + 0.258i)12-s − 1.29i·13-s − 0.979i·14-s + (0.382 + 0.231i)15-s + 0.250·16-s + 0.460i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2010\)    =    \(2 \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.984 + 0.173i$
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: \((2,\ 2010,\ (\ :1/2),\ 0.984 + 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.125530259\)
\(L(\frac12)\) \(\approx\) \(4.125530259\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 - 0.895i)T \)
5 \( 1 - T \)
67 \( 1 + (7.63 - 2.95i)T \)
good7 \( 1 + 3.66iT - 7T^{2} \)
11 \( 1 + 1.69T + 11T^{2} \)
13 \( 1 + 4.67iT - 13T^{2} \)
17 \( 1 - 1.90iT - 17T^{2} \)
19 \( 1 - 5.49T + 19T^{2} \)
23 \( 1 - 2.42iT - 23T^{2} \)
29 \( 1 + 0.513iT - 29T^{2} \)
31 \( 1 + 0.984iT - 31T^{2} \)
37 \( 1 + 1.30T + 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 + 3.48iT - 43T^{2} \)
47 \( 1 - 0.157iT - 47T^{2} \)
53 \( 1 + 5.09T + 53T^{2} \)
59 \( 1 + 6.76iT - 59T^{2} \)
61 \( 1 - 1.01iT - 61T^{2} \)
71 \( 1 - 0.601iT - 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 + 3.27iT - 79T^{2} \)
83 \( 1 - 16.0iT - 83T^{2} \)
89 \( 1 + 10.0iT - 89T^{2} \)
97 \( 1 - 16.9iT - 97T^{2} \)
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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

−9.342019848423900282854690328162, −8.004780003909515115158149907027, −7.73315744050287415810747488477, −6.88621424898521727906927684326, −5.67073250495224879312455536377, −5.04854461527758059382152927116, −4.05134571553418308239931090919, −3.38840138326394202726120013904, −2.55250610196841800206150728040, −1.17712010774523074861862028982, 1.53763078302530027691599687711, 2.49082346137221864497710422919, 2.99491166861728193356149028866, 4.25075459734998874302654269883, 5.21193665131119451694176558377, 5.99529849432723181309516585554, 6.74802244585417194132362275716, 7.54884468873974571119854102143, 8.391040321937367559183880703734, 9.337061530523467502644612430514

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