Properties

Label 2-1840-23.22-c2-0-77
Degree $2$
Conductor $1840$
Sign $0.819 + 0.573i$
Analytic cond. $50.1363$
Root an. cond. $7.08070$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.52·3-s − 2.23i·5-s + 3.78i·7-s + 21.5·9-s − 2.97i·11-s − 0.383·13-s − 12.3i·15-s − 7.29i·17-s − 21.5i·19-s + 20.8i·21-s + (13.1 − 18.8i)23-s − 5.00·25-s + 69.0·27-s − 15.0·29-s + 11.1·31-s + ⋯
L(s)  = 1  + 1.84·3-s − 0.447i·5-s + 0.540i·7-s + 2.38·9-s − 0.270i·11-s − 0.0295·13-s − 0.823i·15-s − 0.428i·17-s − 1.13i·19-s + 0.994i·21-s + (0.573 − 0.819i)23-s − 0.200·25-s + 2.55·27-s − 0.519·29-s + 0.358·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.819 + 0.573i$
Analytic conductor: \(50.1363\)
Root analytic conductor: \(7.08070\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1),\ 0.819 + 0.573i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.348171773\)
\(L(\frac12)\) \(\approx\) \(4.348171773\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + 2.23iT \)
23 \( 1 + (-13.1 + 18.8i)T \)
good3 \( 1 - 5.52T + 9T^{2} \)
7 \( 1 - 3.78iT - 49T^{2} \)
11 \( 1 + 2.97iT - 121T^{2} \)
13 \( 1 + 0.383T + 169T^{2} \)
17 \( 1 + 7.29iT - 289T^{2} \)
19 \( 1 + 21.5iT - 361T^{2} \)
29 \( 1 + 15.0T + 841T^{2} \)
31 \( 1 - 11.1T + 961T^{2} \)
37 \( 1 + 39.8iT - 1.36e3T^{2} \)
41 \( 1 - 53.5T + 1.68e3T^{2} \)
43 \( 1 + 33.1iT - 1.84e3T^{2} \)
47 \( 1 - 10.3T + 2.20e3T^{2} \)
53 \( 1 - 65.9iT - 2.80e3T^{2} \)
59 \( 1 + 10.7T + 3.48e3T^{2} \)
61 \( 1 - 55.7iT - 3.72e3T^{2} \)
67 \( 1 - 102. iT - 4.48e3T^{2} \)
71 \( 1 + 64.5T + 5.04e3T^{2} \)
73 \( 1 - 107.T + 5.32e3T^{2} \)
79 \( 1 + 13.1iT - 6.24e3T^{2} \)
83 \( 1 - 123. iT - 6.88e3T^{2} \)
89 \( 1 + 71.9iT - 7.92e3T^{2} \)
97 \( 1 + 81.7iT - 9.40e3T^{2} \)
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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

−9.048279765096053272140693159760, −8.435350571723132785418845106167, −7.54438755940599106639811616266, −6.97341330359398360985613349900, −5.71493419980337384768846751967, −4.63741036246511882782739106264, −3.89249900580917983928632433800, −2.76447955576158612730251259764, −2.33345912732435484827119338677, −0.943413906198053127241918030875, 1.35872479371920922527173219395, 2.26347371842522641187664818930, 3.30490051261860158498225737902, 3.79934079444435066777260830588, 4.76024246290362915829028181665, 6.15881049255274779542918390117, 7.10020575604916173223651453320, 7.74057989296443135628542133027, 8.215566444668180843057428851055, 9.174110309876906000671724552887

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