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Properties

Label 2-1860-1860.119-c0-0-2
Degree $2$
Conductor $1860$
Sign $0.275 + 0.961i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 0.999·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 − 0.866i)10-s + (−0.499 − 0.866i)12-s + 0.999·15-s + (−0.5 − 0.866i)16-s + (−1.5 − 0.866i)17-s + (0.499 − 0.866i)18-s + (−1.5 − 0.866i)19-s + 0.999·20-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 0.999·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 − 0.866i)10-s + (−0.499 − 0.866i)12-s + 0.999·15-s + (−0.5 − 0.866i)16-s + (−1.5 − 0.866i)17-s + (0.499 − 0.866i)18-s + (−1.5 − 0.866i)19-s + 0.999·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.275 + 0.961i$
Analytic conductor: \(0.928260\)
Root analytic conductor: \(0.963462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :0),\ 0.275 + 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1462083455\)
\(L(\frac12)\) \(\approx\) \(0.1462083455\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - T \)
good7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + 2T + T^{2} \)
29 \( 1 + T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - 1.73iT - T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + 1.73iT - T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.074161476847201786665245082471, −8.533451082316832587064430392932, −7.74811431071225108468416519154, −6.59629879326233210355278359956, −6.09107812417567590873243082713, −5.01158946893955614561058821181, −4.42801517784651418863427104037, −4.00666711565430950102062583216, −2.61006877595871341558296629138, −0.090280572630585089917016260268, 1.89474527404143668492961564638, 2.44701808039901576161071865951, 3.85192981170327075673865920864, 4.38794171526591026738619477018, 5.77267713192476348700344454791, 6.30747225930340887947205421279, 6.94382061982305856191300411782, 8.208332825457758440542213104709, 8.556601155658246301973932538455, 10.10823562432668908806115536355

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