Properties

Label 2-1840-23.22-c2-0-81
Degree $2$
Conductor $1840$
Sign $0.0838 + 0.996i$
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  + 4.25·3-s − 2.23i·5-s − 7.64i·7-s + 9.06·9-s − 8.27i·11-s + 17.8·13-s − 9.50i·15-s − 17.4i·17-s + 21.3i·19-s − 32.4i·21-s + (22.9 − 1.92i)23-s − 5.00·25-s + 0.277·27-s − 17.3·29-s + 16.3·31-s + ⋯
L(s)  = 1  + 1.41·3-s − 0.447i·5-s − 1.09i·7-s + 1.00·9-s − 0.752i·11-s + 1.37·13-s − 0.633i·15-s − 1.02i·17-s + 1.12i·19-s − 1.54i·21-s + (0.996 − 0.0838i)23-s − 0.200·25-s + 0.0102·27-s − 0.597·29-s + 0.526·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0838 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0838 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.0838 + 0.996i$
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: \((2,\ 1840,\ (\ :1),\ 0.0838 + 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.544624445\)
\(L(\frac12)\) \(\approx\) \(3.544624445\)
\(L(2)\) 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 \( 1 + (-22.9 + 1.92i)T \)
good3 \( 1 - 4.25T + 9T^{2} \)
7 \( 1 + 7.64iT - 49T^{2} \)
11 \( 1 + 8.27iT - 121T^{2} \)
13 \( 1 - 17.8T + 169T^{2} \)
17 \( 1 + 17.4iT - 289T^{2} \)
19 \( 1 - 21.3iT - 361T^{2} \)
29 \( 1 + 17.3T + 841T^{2} \)
31 \( 1 - 16.3T + 961T^{2} \)
37 \( 1 - 20.0iT - 1.36e3T^{2} \)
41 \( 1 - 1.54T + 1.68e3T^{2} \)
43 \( 1 + 54.5iT - 1.84e3T^{2} \)
47 \( 1 + 88.4T + 2.20e3T^{2} \)
53 \( 1 + 37.7iT - 2.80e3T^{2} \)
59 \( 1 + 49.0T + 3.48e3T^{2} \)
61 \( 1 - 88.6iT - 3.72e3T^{2} \)
67 \( 1 + 14.0iT - 4.48e3T^{2} \)
71 \( 1 - 113.T + 5.04e3T^{2} \)
73 \( 1 - 41.3T + 5.32e3T^{2} \)
79 \( 1 + 64.3iT - 6.24e3T^{2} \)
83 \( 1 + 147. iT - 6.88e3T^{2} \)
89 \( 1 + 38.4iT - 7.92e3T^{2} \)
97 \( 1 + 74.4iT - 9.40e3T^{2} \)
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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

−8.737679140306614667175298127586, −8.239907673406153708522380276089, −7.54633301694308302156036622556, −6.68836765282212745400723567682, −5.64924772837231308410737389340, −4.51138831420177266986045127817, −3.58361007442240707508572892483, −3.17913129536854189417259718306, −1.74389205362279020383303125116, −0.76348432159491068781470619677, 1.54356337018367575335780559087, 2.45477033769168533478073084538, 3.18174764849812418168522453141, 4.02041679165329444248453362613, 5.15349317416329912087103228741, 6.22785164404899317076037730600, 6.91259653037468453235202621048, 8.051509648933935800429108588645, 8.366368035462957828163752286472, 9.338791580848217234250595095023

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