↑

Properties

Label 2-1860-155.92-c1-0-18
Degree $2$
Conductor $1860$
Sign $0.984 + 0.177i$
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  + (−0.707 − 0.707i)3-s + (1.82 + 1.29i)5-s + (2.05 + 2.05i)7-s + 1.00i·9-s − 1.00i·11-s + (−1.61 − 1.61i)13-s + (−0.374 − 2.20i)15-s + (3.86 − 3.86i)17-s − 5.41i·19-s − 2.91i·21-s + (−0.0230 − 0.0230i)23-s + (1.64 + 4.72i)25-s + (0.707 − 0.707i)27-s − 1.52·29-s + (4.36 − 3.46i)31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.815 + 0.578i)5-s + (0.777 + 0.777i)7-s + 0.333i·9-s − 0.303i·11-s + (−0.449 − 0.449i)13-s + (−0.0966 − 0.569i)15-s + (0.937 − 0.937i)17-s − 1.24i·19-s − 0.635i·21-s + (−0.00480 − 0.00480i)23-s + (0.329 + 0.944i)25-s + (0.136 − 0.136i)27-s − 0.282·29-s + (0.783 − 0.621i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.984 + 0.177i$
Analytic conductor: \(14.8521\)
Root analytic conductor: \(3.85385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (1177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :1/2),\ 0.984 + 0.177i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.980092660\)
\(L(\frac12)\) \(\approx\) \(1.980092660\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-1.82 - 1.29i)T \)
31 \( 1 + (-4.36 + 3.46i)T \)
good7 \( 1 + (-2.05 - 2.05i)T + 7iT^{2} \)
11 \( 1 + 1.00iT - 11T^{2} \)
13 \( 1 + (1.61 + 1.61i)T + 13iT^{2} \)
17 \( 1 + (-3.86 + 3.86i)T - 17iT^{2} \)
19 \( 1 + 5.41iT - 19T^{2} \)
23 \( 1 + (0.0230 + 0.0230i)T + 23iT^{2} \)
29 \( 1 + 1.52T + 29T^{2} \)
37 \( 1 + (-3.04 + 3.04i)T - 37iT^{2} \)
41 \( 1 - 4.90T + 41T^{2} \)
43 \( 1 + (-3.37 - 3.37i)T + 43iT^{2} \)
47 \( 1 + (6.30 + 6.30i)T + 47iT^{2} \)
53 \( 1 + (-8.56 - 8.56i)T + 53iT^{2} \)
59 \( 1 - 8.14iT - 59T^{2} \)
61 \( 1 - 14.6iT - 61T^{2} \)
67 \( 1 + (-5.16 - 5.16i)T + 67iT^{2} \)
71 \( 1 - 8.00T + 71T^{2} \)
73 \( 1 + (-4.34 - 4.34i)T + 73iT^{2} \)
79 \( 1 + 3.07T + 79T^{2} \)
83 \( 1 + (10.8 + 10.8i)T + 83iT^{2} \)
89 \( 1 + 2.30T + 89T^{2} \)
97 \( 1 + (-1.43 - 1.43i)T + 97iT^{2} \)
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

−9.276684719465418132940267866952, −8.416834434273450611959820489692, −7.48569566351369999064801262718, −6.92028856469028282747540192339, −5.72442774838071433155986323347, −5.53543204380785032126611733646, −4.50160913908671617844677567550, −2.86254252205477904994978886731, −2.35987945555593245128087682362, −0.971920454755195091621359834038, 1.10826053655225510852317496239, 2.00050872536656648943428111396, 3.59851373855915185262992668411, 4.46920407228350435291896762845, 5.14251010037271595492501449843, 5.95012275713965414777621483640, 6.75824503151779084328365817640, 7.890286472377260798188112412326, 8.347546259319354310353487002766, 9.602260151817701094842407620773

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