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Properties

Label 2-2040-2040.1589-c0-0-4
Degree $2$
Conductor $2040$
Sign $0.997 - 0.0758i$
Analytic cond. $1.01809$
Root an. cond. $1.00900$
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  + (0.923 − 0.382i)2-s + (0.923 + 0.382i)3-s + (0.707 − 0.707i)4-s + (−0.382 + 0.923i)5-s + 6-s + (0.382 − 0.923i)8-s + (0.707 + 0.707i)9-s + i·10-s + (0.923 − 0.382i)12-s + (−0.707 + 0.707i)15-s − i·16-s + (−0.923 − 0.382i)17-s + (0.923 + 0.382i)18-s + (1 + i)19-s + (0.382 + 0.923i)20-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)2-s + (0.923 + 0.382i)3-s + (0.707 − 0.707i)4-s + (−0.382 + 0.923i)5-s + 6-s + (0.382 − 0.923i)8-s + (0.707 + 0.707i)9-s + i·10-s + (0.923 − 0.382i)12-s + (−0.707 + 0.707i)15-s − i·16-s + (−0.923 − 0.382i)17-s + (0.923 + 0.382i)18-s + (1 + i)19-s + (0.382 + 0.923i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2040\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.997 - 0.0758i$
Analytic conductor: \(1.01809\)
Root analytic conductor: \(1.00900\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2040} (1589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2040,\ (\ :0),\ 0.997 - 0.0758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.574304365\)
\(L(\frac12)\) \(\approx\) \(2.574304365\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.923 + 0.382i)T \)
3 \( 1 + (-0.923 - 0.382i)T \)
5 \( 1 + (0.382 - 0.923i)T \)
17 \( 1 + (0.923 + 0.382i)T \)
good7 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 + (0.765 + 1.84i)T + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.707 + 0.707i)T^{2} \)
31 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (-0.707 + 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - 0.765iT - T^{2} \)
53 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
83 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.707 - 0.707i)T^{2} \)
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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

−9.500315614655491883752499839084, −8.550642108342979475974304857421, −7.65282060390917182045679009808, −6.95014133236166279994565148070, −6.21182772926104706700934579006, −5.03978473647593458773544801749, −4.24034353372288347153749881125, −3.46340276503039209821127577633, −2.77178331893976818226220870978, −1.86397135090828589264416988856, 1.57118634176543465745634873669, 2.62894007798636017864219017630, 3.74209048962161599954078629126, 4.24238802340066628613892124691, 5.28583988987904774224288417733, 6.06376499964163917147819070276, 7.23210762264174864704413170368, 7.57417681887513041332556155568, 8.384331686223679899187058383463, 9.115794931053844400078873542328

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