Properties

Label 2-1860-465.464-c1-0-37
Degree $2$
Conductor $1860$
Sign $0.910 + 0.413i$
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  + (−1.62 + 0.599i)3-s + (2.11 − 0.714i)5-s − 1.63i·7-s + (2.28 − 1.94i)9-s + 2.01·11-s + 1.37·13-s + (−3.01 + 2.43i)15-s + 3.11i·17-s + 1.91·19-s + (0.982 + 2.66i)21-s + 2.43i·23-s + (3.97 − 3.02i)25-s + (−2.53 + 4.53i)27-s + 3.79·29-s + (−2.50 − 4.97i)31-s + ⋯
L(s)  = 1  + (−0.938 + 0.346i)3-s + (0.947 − 0.319i)5-s − 0.619i·7-s + (0.760 − 0.649i)9-s + 0.608·11-s + 0.381·13-s + (−0.778 + 0.628i)15-s + 0.754i·17-s + 0.439·19-s + (0.214 + 0.580i)21-s + 0.506i·23-s + (0.795 − 0.605i)25-s + (−0.488 + 0.872i)27-s + 0.704·29-s + (−0.449 − 0.893i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.910 + 0.413i$
Analytic conductor: \(14.8521\)
Root analytic conductor: \(3.85385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :1/2),\ 0.910 + 0.413i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.636486577\)
\(L(\frac12)\) \(\approx\) \(1.636486577\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 0.599i)T \)
5 \( 1 + (-2.11 + 0.714i)T \)
31 \( 1 + (2.50 + 4.97i)T \)
good7 \( 1 + 1.63iT - 7T^{2} \)
11 \( 1 - 2.01T + 11T^{2} \)
13 \( 1 - 1.37T + 13T^{2} \)
17 \( 1 - 3.11iT - 17T^{2} \)
19 \( 1 - 1.91T + 19T^{2} \)
23 \( 1 - 2.43iT - 23T^{2} \)
29 \( 1 - 3.79T + 29T^{2} \)
37 \( 1 + 1.29T + 37T^{2} \)
41 \( 1 + 8.16iT - 41T^{2} \)
43 \( 1 + 6.33T + 43T^{2} \)
47 \( 1 - 1.24T + 47T^{2} \)
53 \( 1 + 4.51iT - 53T^{2} \)
59 \( 1 + 5.69iT - 59T^{2} \)
61 \( 1 - 6.88iT - 61T^{2} \)
67 \( 1 - 13.2iT - 67T^{2} \)
71 \( 1 - 3.93iT - 71T^{2} \)
73 \( 1 - 3.75T + 73T^{2} \)
79 \( 1 + 9.55iT - 79T^{2} \)
83 \( 1 + 6.14iT - 83T^{2} \)
89 \( 1 - 16.3T + 89T^{2} \)
97 \( 1 - 7.01iT - 97T^{2} \)
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

−9.307544435049215659723637508492, −8.618358520334654367617479924177, −7.40719174982824504951879250254, −6.61852732654155481591137844242, −5.93010151223807110490019052831, −5.26244846788078407500167300832, −4.30115584041957545661517310067, −3.53215960472109001216603514747, −1.85868146308056669094816047168, −0.853188025592237283718851717660, 1.10232348080724674745163520896, 2.14649670780413804275063702969, 3.25321095280979071476743603371, 4.71175846743883521631359659550, 5.33084140862306119553921334394, 6.22765985136848080032723082492, 6.62560773978240383356415892432, 7.51544150314401238274823000548, 8.626520632449566977646906393367, 9.376742240044471999415317321909

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