Properties

Label 2-1840-23.22-c2-0-57
Degree $2$
Conductor $1840$
Sign $0.626 + 0.779i$
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  − 1.59·3-s + 2.23i·5-s + 7.14i·7-s − 6.44·9-s − 5.16i·11-s + 13.8·13-s − 3.57i·15-s − 15.0i·17-s − 5.66i·19-s − 11.4i·21-s + (−17.9 + 14.4i)23-s − 5.00·25-s + 24.6·27-s − 13.7·29-s − 57.6·31-s + ⋯
L(s)  = 1  − 0.532·3-s + 0.447i·5-s + 1.02i·7-s − 0.716·9-s − 0.469i·11-s + 1.06·13-s − 0.238i·15-s − 0.887i·17-s − 0.298i·19-s − 0.543i·21-s + (−0.779 + 0.626i)23-s − 0.200·25-s + 0.914·27-s − 0.474·29-s − 1.85·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.626 + 0.779i$
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: \((2,\ 1840,\ (\ :1),\ 0.626 + 0.779i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9820279780\)
\(L(\frac12)\) \(\approx\) \(0.9820279780\)
\(L(2)\) 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 \( 1 + (17.9 - 14.4i)T \)
good3 \( 1 + 1.59T + 9T^{2} \)
7 \( 1 - 7.14iT - 49T^{2} \)
11 \( 1 + 5.16iT - 121T^{2} \)
13 \( 1 - 13.8T + 169T^{2} \)
17 \( 1 + 15.0iT - 289T^{2} \)
19 \( 1 + 5.66iT - 361T^{2} \)
29 \( 1 + 13.7T + 841T^{2} \)
31 \( 1 + 57.6T + 961T^{2} \)
37 \( 1 + 49.1iT - 1.36e3T^{2} \)
41 \( 1 - 12.0T + 1.68e3T^{2} \)
43 \( 1 - 61.9iT - 1.84e3T^{2} \)
47 \( 1 + 1.87T + 2.20e3T^{2} \)
53 \( 1 + 77.3iT - 2.80e3T^{2} \)
59 \( 1 + 4.88T + 3.48e3T^{2} \)
61 \( 1 - 18.9iT - 3.72e3T^{2} \)
67 \( 1 - 30.5iT - 4.48e3T^{2} \)
71 \( 1 - 55.5T + 5.04e3T^{2} \)
73 \( 1 - 110.T + 5.32e3T^{2} \)
79 \( 1 + 79.2iT - 6.24e3T^{2} \)
83 \( 1 + 38.5iT - 6.88e3T^{2} \)
89 \( 1 + 22.3iT - 7.92e3T^{2} \)
97 \( 1 - 16.8iT - 9.40e3T^{2} \)
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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

−9.008734210937852918689937660073, −8.253475498339287051356059900009, −7.34455032168324866423238749352, −6.33412965831780700690127079299, −5.73296401249408822557072634797, −5.23026413842410129343247373536, −3.81578304503429426380020243485, −2.98167288648371035615944807775, −1.95051364924412652881025282818, −0.34844366482083653704469109606, 0.869366690581302985858309749205, 1.97727640090768819496683453291, 3.56053710539557693661343026494, 4.13854324121209057715159801320, 5.20991744460757015230425799431, 5.97000688342972544909024067052, 6.67862217979619064390583940771, 7.66449411595429445969330188318, 8.381380262668168723150595514963, 9.097847688090471924624125943090

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