↑

Properties

Label 2-1860-31.5-c1-0-18
Degree $2$
Conductor $1860$
Sign $-0.640 + 0.767i$
Analytic cond. $14.8521$
Root an. cond. $3.85385$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (1.87 − 3.24i)7-s + (−0.499 − 0.866i)9-s + (−2.24 − 3.89i)11-s + (2.50 + 4.33i)13-s − 0.999·15-s + (−1.17 + 2.02i)17-s + (3.50 − 6.07i)19-s + (−1.87 − 3.24i)21-s + 6.06·23-s + (−0.499 + 0.866i)25-s − 0.999·27-s − 8.35·29-s + (1.61 − 5.32i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.223 − 0.387i)5-s + (0.708 − 1.22i)7-s + (−0.166 − 0.288i)9-s + (−0.677 − 1.17i)11-s + (0.694 + 1.20i)13-s − 0.258·15-s + (−0.284 + 0.492i)17-s + (0.804 − 1.39i)19-s + (−0.408 − 0.708i)21-s + 1.26·23-s + (−0.0999 + 0.173i)25-s − 0.192·27-s − 1.55·29-s + (0.290 − 0.956i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.640 + 0.767i$
Analytic conductor: \(14.8521\)
Root analytic conductor: \(3.85385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (1741, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :1/2),\ -0.640 + 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.749928235\)
\(L(\frac12)\) \(\approx\) \(1.749928235\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-1.61 + 5.32i)T \)
good7 \( 1 + (-1.87 + 3.24i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.24 + 3.89i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.50 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.17 - 2.02i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.50 + 6.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.06T + 23T^{2} \)
29 \( 1 + 8.35T + 29T^{2} \)
37 \( 1 + (-0.873 + 1.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.94 - 3.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.931 + 1.61i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.40T + 47T^{2} \)
53 \( 1 + (2.90 + 5.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.05 - 7.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 13.3T + 61T^{2} \)
67 \( 1 + (1.34 + 2.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.96 + 5.14i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-8.20 - 14.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.16 - 7.21i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.16 + 3.74i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 - 4.31T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.904706769299123467472168107333, −8.091480931682487848482024259010, −7.45935454312569515987436335840, −6.77339303159649095576605325331, −5.76645365762442911298135567983, −4.74930275350126113196396250274, −3.97713408972867645139903902802, −2.99894734516263879892210494603, −1.61041966553173728845021799974, −0.63673578120553434146136664896, 1.68933925057077363732023412066, 2.77363253219300097283075040193, 3.50127211064877830272822943334, 4.84053574063122196040193332512, 5.30209491912078637966448466705, 6.16835479898998344244660638063, 7.57027276249759898737378311406, 7.80601006759255550022263508633, 8.820108531767840691659302398853, 9.416021665487119635657487174918

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