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Properties

Label 2-1824-76.27-c1-0-12
Degree $2$
Conductor $1824$
Sign $-0.783 + 0.621i$
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 + (−1.46 + 2.53i)5-s + 3.81i·7-s + (−0.499 − 0.866i)9-s + 4.60i·11-s + (−2.57 + 1.48i)13-s + (−1.46 − 2.53i)15-s + (−3.87 + 6.70i)17-s + (−4.26 − 0.919i)19-s + (−3.30 − 1.90i)21-s + (7.85 − 4.53i)23-s + (−1.79 − 3.10i)25-s + 0.999·27-s + (2.34 − 1.35i)29-s + 1.07·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.655 + 1.13i)5-s + 1.44i·7-s + (−0.166 − 0.288i)9-s + 1.38i·11-s + (−0.713 + 0.412i)13-s + (−0.378 − 0.655i)15-s + (−0.938 + 1.62i)17-s + (−0.977 − 0.211i)19-s + (−0.720 − 0.416i)21-s + (1.63 − 0.945i)23-s + (−0.358 − 0.620i)25-s + 0.192·27-s + (0.434 − 0.251i)29-s + 0.192·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1824\)    =    \(2^{5} \cdot 3 \cdot 19\)
Sign: $-0.783 + 0.621i$
Analytic conductor: \(14.5647\)
Root analytic conductor: \(3.81637\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1824} (1471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1824,\ (\ :1/2),\ -0.783 + 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9575229375\)
\(L(\frac12)\) \(\approx\) \(0.9575229375\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.26 + 0.919i)T \)
good5 \( 1 + (1.46 - 2.53i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.81iT - 7T^{2} \)
11 \( 1 - 4.60iT - 11T^{2} \)
13 \( 1 + (2.57 - 1.48i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.87 - 6.70i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-7.85 + 4.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.34 + 1.35i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.07T + 31T^{2} \)
37 \( 1 + 0.681iT - 37T^{2} \)
41 \( 1 + (-6.78 - 3.91i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.42 - 1.40i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-9.37 + 5.41i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.25 - 0.722i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.324 - 0.561i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.62 - 9.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.28 - 3.96i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.80 + 10.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.28 - 5.68i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.259 - 0.449i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.15iT - 83T^{2} \)
89 \( 1 + (3.43 - 1.98i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.38 + 2.53i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.749942420922105480560454626886, −8.943802465888316982252900534100, −8.327836881208047927755538858733, −7.12455635087613479403074066006, −6.66034930901523018610107388727, −5.80060253974315190120051481820, −4.64553296653382290575407160837, −4.13867925040862603212831987358, −2.73684078620429440859546734830, −2.20625723666037852812346445656, 0.46741479766044219904322404773, 0.951937936457600914757168956086, 2.73822202223287295028446188535, 3.87664992884782645026642887726, 4.72195795425045182761265432231, 5.35262613302079210061332724445, 6.56737105571530833581689599756, 7.31104886602584356364616164071, 7.86746845678122861150163028660, 8.762652160639935009392935401432

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