↑

Properties

Label 2-1860-1860.1139-c0-0-2
Degree $2$
Conductor $1860$
Sign $0.965 - 0.258i$
Analytic cond. $0.928260$
Root an. cond. $0.963462$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + 5-s + (0.809 − 0.587i)6-s + (0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.309 − 0.951i)10-s + (−0.809 − 0.587i)12-s + (0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (0.809 + 0.587i)18-s + (1.80 + 0.587i)19-s + (−0.809 + 0.587i)20-s + (−1.30 + 0.951i)23-s + (−0.309 + 0.951i)24-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + 5-s + (0.809 − 0.587i)6-s + (0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.309 − 0.951i)10-s + (−0.809 − 0.587i)12-s + (0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (0.809 + 0.587i)18-s + (1.80 + 0.587i)19-s + (−0.809 + 0.587i)20-s + (−1.30 + 0.951i)23-s + (−0.309 + 0.951i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.965 - 0.258i$
Analytic conductor: \(0.928260\)
Root analytic conductor: \(0.963462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :0),\ 0.965 - 0.258i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.179786412\)
\(L(\frac12)\) \(\approx\) \(1.179786412\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 - T \)
31 \( 1 + (-0.309 + 0.951i)T \)
good7 \( 1 + (0.309 + 0.951i)T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
29 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (0.809 + 0.587i)T^{2} \)
47 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T^{2} \)
61 \( 1 - 1.90iT - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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

−9.681314105358588707570906243937, −9.063560436350543976633838790193, −8.162050826807470876291038056355, −7.43876006367582684911582931766, −5.87162349499340503398493004097, −5.33982100999309271339809561532, −4.32912171660470875000827340324, −3.44343095677561847994309735259, −2.61883253884069359914216627617, −1.54632828859088699776456699409, 1.06947050487042783174678019421, 2.20519177403856578838694642964, 3.43040812018609698711606768303, 4.92870687516810184692643415484, 5.57279402041584039770696279142, 6.49119982247709318411973322592, 6.86117299035637665850889405461, 7.86683741671395288550662338482, 8.414033423555046015373566840060, 9.324599310834984565141929940022

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