Properties

Label 2-1840-23.22-c2-0-29
Degree $2$
Conductor $1840$
Sign $0.999 + 0.0321i$
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  − 3.98·3-s − 2.23i·5-s − 6.68i·7-s + 6.91·9-s + 10.4i·11-s − 4.43·13-s + 8.92i·15-s − 1.02i·17-s − 24.8i·19-s + 26.6i·21-s + (−0.739 + 22.9i)23-s − 5.00·25-s + 8.32·27-s + 20.9·29-s − 35.4·31-s + ⋯
L(s)  = 1  − 1.32·3-s − 0.447i·5-s − 0.954i·7-s + 0.768·9-s + 0.952i·11-s − 0.340·13-s + 0.594i·15-s − 0.0605i·17-s − 1.30i·19-s + 1.26i·21-s + (−0.0321 + 0.999i)23-s − 0.200·25-s + 0.308·27-s + 0.720·29-s − 1.14·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0321i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.999 + 0.0321i$
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: \((2,\ 1840,\ (\ :1),\ 0.999 + 0.0321i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8254515454\)
\(L(\frac12)\) \(\approx\) \(0.8254515454\)
\(L(2)\) 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 \( 1 + (0.739 - 22.9i)T \)
good3 \( 1 + 3.98T + 9T^{2} \)
7 \( 1 + 6.68iT - 49T^{2} \)
11 \( 1 - 10.4iT - 121T^{2} \)
13 \( 1 + 4.43T + 169T^{2} \)
17 \( 1 + 1.02iT - 289T^{2} \)
19 \( 1 + 24.8iT - 361T^{2} \)
29 \( 1 - 20.9T + 841T^{2} \)
31 \( 1 + 35.4T + 961T^{2} \)
37 \( 1 - 50.9iT - 1.36e3T^{2} \)
41 \( 1 + 53.5T + 1.68e3T^{2} \)
43 \( 1 - 64.2iT - 1.84e3T^{2} \)
47 \( 1 - 2.03T + 2.20e3T^{2} \)
53 \( 1 - 13.7iT - 2.80e3T^{2} \)
59 \( 1 + 7.09T + 3.48e3T^{2} \)
61 \( 1 + 68.5iT - 3.72e3T^{2} \)
67 \( 1 + 80.9iT - 4.48e3T^{2} \)
71 \( 1 + 117.T + 5.04e3T^{2} \)
73 \( 1 - 38.6T + 5.32e3T^{2} \)
79 \( 1 - 11.9iT - 6.24e3T^{2} \)
83 \( 1 - 10.8iT - 6.88e3T^{2} \)
89 \( 1 - 17.7iT - 7.92e3T^{2} \)
97 \( 1 - 5.84iT - 9.40e3T^{2} \)
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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

−9.293799725832137236049143660249, −8.151979616476276099794329804994, −7.19652105872291686788244438628, −6.78191410022176819636437815794, −5.79471793663781218864509070011, −4.73929097010987268048300558181, −4.65872411284368760748056062265, −3.23031258792256590536690219693, −1.66622224330387636413936419062, −0.60241406234677434875256181431, 0.46215906559862600960055601939, 1.97487783148514026825273432879, 3.11718762760679874638664405897, 4.20895122758458272592935915384, 5.47083566097024969193335601009, 5.67606541311636708771739615878, 6.47787108627984005279163087939, 7.29027176464228230360617878828, 8.410133552435142549434864713188, 8.960055109430326073536345663406

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