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Properties

Label 2-1824-76.31-c1-0-16
Degree $2$
Conductor $1824$
Sign $0.985 + 0.167i$
Analytic cond. $14.5647$
Root an. cond. $3.81637$
Motivic weight $1$
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)3-s + (0.508 + 0.880i)5-s + 1.27i·7-s + (−0.499 + 0.866i)9-s + 0.0879i·11-s + (−3.47 − 2.00i)13-s + (0.508 − 0.880i)15-s + (−2.85 − 4.94i)17-s + (2.85 + 3.29i)19-s + (1.10 − 0.637i)21-s + (3.99 + 2.30i)23-s + (1.98 − 3.43i)25-s + 0.999·27-s + (2.52 + 1.45i)29-s + 7.52·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.227 + 0.393i)5-s + 0.481i·7-s + (−0.166 + 0.288i)9-s + 0.0265i·11-s + (−0.962 − 0.555i)13-s + (0.131 − 0.227i)15-s + (−0.692 − 1.20i)17-s + (0.655 + 0.754i)19-s + (0.240 − 0.139i)21-s + (0.832 + 0.480i)23-s + (0.396 − 0.686i)25-s + 0.192·27-s + (0.468 + 0.270i)29-s + 1.35·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1824\)    =    \(2^{5} \cdot 3 \cdot 19\)
Sign: $0.985 + 0.167i$
Analytic conductor: \(14.5647\)
Root analytic conductor: \(3.81637\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1824} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1824,\ (\ :1/2),\ 0.985 + 0.167i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.532552957\)
\(L(\frac12)\) \(\approx\) \(1.532552957\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-2.85 - 3.29i)T \)
good5 \( 1 + (-0.508 - 0.880i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.27iT - 7T^{2} \)
11 \( 1 - 0.0879iT - 11T^{2} \)
13 \( 1 + (3.47 + 2.00i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.85 + 4.94i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.99 - 2.30i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.52 - 1.45i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.52T + 31T^{2} \)
37 \( 1 + 2.94iT - 37T^{2} \)
41 \( 1 + (-2.94 + 1.70i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.88 + 1.09i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.26 - 1.30i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.74 - 2.74i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.68 - 2.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.37 - 11.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.40 - 4.16i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.644 - 1.11i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.09 - 5.35i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.52 - 13.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.0iT - 83T^{2} \)
89 \( 1 + (-3.84 - 2.21i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.08 - 4.66i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.241915377013628323085390451479, −8.444881567894864971138629191712, −7.41591846217275870976285849452, −7.02823075217146049578008872078, −5.99717364599742892750458568434, −5.31647447865254747637086994675, −4.45797905942312061921559096819, −2.92600929853493127829855655923, −2.42021300084853458934592320246, −0.859411671960203689294562887336, 0.861707482474910977157648767490, 2.30551693892863636395087811584, 3.46811959375520683565519720798, 4.64749752181294589369826142638, 4.87109127350165769005184836145, 6.13198856660395782522151785914, 6.80712765389196473894523009974, 7.68987588798838824086860138783, 8.708588126085141985910740112270, 9.274640884702503962079735818980

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