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Properties

Label 2-1872-52.23-c0-0-2
Degree $2$
Conductor $1872$
Sign $0.859 + 0.511i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
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)7-s + (−0.5 + 0.866i)13-s + (1 − 1.73i)19-s + 25-s + 31-s + (−1.5 − 0.866i)43-s + (−0.5 + 0.866i)61-s + (0.5 + 0.866i)67-s − 1.73i·73-s − 1.73i·79-s + (0.499 + 0.866i)91-s + (1.5 + 0.866i)97-s + 1.73i·103-s + 1.73i·109-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)13-s + (1 − 1.73i)19-s + 25-s + 31-s + (−1.5 − 0.866i)43-s + (−0.5 + 0.866i)61-s + (0.5 + 0.866i)67-s − 1.73i·73-s − 1.73i·79-s + (0.499 + 0.866i)91-s + (1.5 + 0.866i)97-s + 1.73i·103-s + 1.73i·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :0),\ 0.859 + 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.215742933\)
\(L(\frac12)\) \(\approx\) \(1.215742933\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 - T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 + 1.73iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.270122457354307106114620230900, −8.688493767395100143807968470441, −7.59900747279923966940697472165, −7.09541984055325404482419459072, −6.36324519490532612749923817031, −4.95094137534503040634238591449, −4.68298980380325634009599432509, −3.48949418039190592007887775730, −2.41945539463594212418224734452, −1.06138918427718956768636872321, 1.42447038187771156563102882191, 2.64956560146176588859850882204, 3.50350428789934877603123737609, 4.80047423870362881622086045835, 5.41385126673182048669050671591, 6.18205017944184376491011712210, 7.21578883377869743361612671285, 8.151379180294265031923847796239, 8.438880104983498679039953818229, 9.653846259463153295125125448199

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