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Properties

Label 2-2040-2040.1739-c0-0-3
Degree $2$
Conductor $2040$
Sign $0.813 - 0.581i$
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  + (−0.195 + 0.980i)2-s + (−0.555 − 0.831i)3-s + (−0.923 − 0.382i)4-s + (0.980 + 0.195i)5-s + (0.923 − 0.382i)6-s + (0.555 − 0.831i)8-s + (−0.382 + 0.923i)9-s + (−0.382 + 0.923i)10-s + (0.195 + 0.980i)12-s + (−0.382 − 0.923i)15-s + (0.707 + 0.707i)16-s + (0.831 − 0.555i)17-s + (−0.831 − 0.555i)18-s + (0.707 + 1.70i)19-s + (−0.831 − 0.555i)20-s + ⋯
L(s)  = 1  + (−0.195 + 0.980i)2-s + (−0.555 − 0.831i)3-s + (−0.923 − 0.382i)4-s + (0.980 + 0.195i)5-s + (0.923 − 0.382i)6-s + (0.555 − 0.831i)8-s + (−0.382 + 0.923i)9-s + (−0.382 + 0.923i)10-s + (0.195 + 0.980i)12-s + (−0.382 − 0.923i)15-s + (0.707 + 0.707i)16-s + (0.831 − 0.555i)17-s + (−0.831 − 0.555i)18-s + (0.707 + 1.70i)19-s + (−0.831 − 0.555i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2040\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.813 - 0.581i$
Analytic conductor: \(1.01809\)
Root analytic conductor: \(1.00900\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2040} (1739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2040,\ (\ :0),\ 0.813 - 0.581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9622017342\)
\(L(\frac12)\) \(\approx\) \(0.9622017342\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.195 - 0.980i)T \)
3 \( 1 + (0.555 + 0.831i)T \)
5 \( 1 + (-0.980 - 0.195i)T \)
17 \( 1 + (-0.831 + 0.555i)T \)
good7 \( 1 + (-0.923 + 0.382i)T^{2} \)
11 \( 1 + (-0.382 - 0.923i)T^{2} \)
13 \( 1 + iT^{2} \)
19 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (1.17 + 0.785i)T + (0.382 + 0.923i)T^{2} \)
29 \( 1 + (0.923 + 0.382i)T^{2} \)
31 \( 1 + (0.216 + 0.324i)T + (-0.382 + 0.923i)T^{2} \)
37 \( 1 + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (0.923 - 0.382i)T^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 + (-0.275 + 0.275i)T - iT^{2} \)
53 \( 1 + (-1.53 + 0.636i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.382 - 1.92i)T + (-0.923 + 0.382i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.382 + 0.923i)T^{2} \)
73 \( 1 + (0.923 + 0.382i)T^{2} \)
79 \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \)
83 \( 1 + (-1.02 + 0.425i)T + (0.707 - 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (-0.923 - 0.382i)T^{2} \)
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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

−9.314101672389369706838927416399, −8.390350856161041338250674035016, −7.68122086067436058139721442229, −7.05088231854258533114013384197, −6.13884652166056105359033571546, −5.73041195860306994304021019104, −5.07800168803896162542275715179, −3.77215033442458640817958582269, −2.26571631139637503912308136098, −1.12477480230282864938890650023, 1.06021823057225012256839029309, 2.38600587660272000438653711816, 3.38796908591682794739069125785, 4.28560224975506130786063628125, 5.25006877521197267534739743200, 5.64674454682969056385305183540, 6.78953746378828222949886278085, 7.969941289255585633392346464558, 8.947078891502768467646906124393, 9.457203659753930641895342374497

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