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Properties

Label 2-1539-171.94-c0-0-3
Degree $2$
Conductor $1539$
Sign $0.939 + 0.342i$
Analytic cond. $0.768061$
Root an. cond. $0.876390$
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  + (−0.5 + 0.866i)4-s + (1 − 1.73i)5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)11-s + (−0.499 − 0.866i)16-s + 17-s + 19-s + (0.999 + 1.73i)20-s + (−0.5 + 0.866i)23-s + (−1.49 − 2.59i)25-s − 0.999·28-s + 1.99·35-s + (0.5 + 0.866i)43-s + 0.999·44-s + (−0.5 − 0.866i)47-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + (1 − 1.73i)5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)11-s + (−0.499 − 0.866i)16-s + 17-s + 19-s + (0.999 + 1.73i)20-s + (−0.5 + 0.866i)23-s + (−1.49 − 2.59i)25-s − 0.999·28-s + 1.99·35-s + (0.5 + 0.866i)43-s + 0.999·44-s + (−0.5 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1539\)    =    \(3^{4} \cdot 19\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(0.768061\)
Root analytic conductor: \(0.876390\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1539} (1405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1539,\ (\ :0),\ 0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.229429079\)
\(L(\frac12)\) \(\approx\) \(1.229429079\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 - T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.473350366130719601899813161456, −8.688641296225293314978865255507, −8.282249022368222087371795885205, −7.57377715365348270152309666477, −5.91562007881711043693400735385, −5.40007072398068789325028412127, −4.85292776070008308252552085477, −3.66804513659111975324488354019, −2.48586216376119661316447639978, −1.19677184367361732318077074642, 1.51032557721313214793448982898, 2.53988542889064469549143382216, 3.68863392047217206878572793539, 4.84111666168071033696443910353, 5.66063028445808983305019119877, 6.41565335976116401627386509961, 7.25223459027173445593860181239, 7.82044468243207811765996315903, 9.267016579407686003073661140496, 9.955146669020100099234226323524

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