Properties

Label 2-1840-23.22-c2-0-63
Degree $2$
Conductor $1840$
Sign $0.551 + 0.833i$
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  + 1.73·3-s − 2.23i·5-s − 1.02i·7-s − 6.00·9-s + 18.8i·11-s + 0.710·13-s − 3.86i·15-s − 6.02i·17-s − 22.2i·19-s − 1.77i·21-s + (19.1 − 12.6i)23-s − 5.00·25-s − 25.9·27-s + 22.1·29-s − 5.70·31-s + ⋯
L(s)  = 1  + 0.576·3-s − 0.447i·5-s − 0.146i·7-s − 0.667·9-s + 1.70i·11-s + 0.0546·13-s − 0.257i·15-s − 0.354i·17-s − 1.17i·19-s − 0.0844i·21-s + (0.833 − 0.551i)23-s − 0.200·25-s − 0.961·27-s + 0.765·29-s − 0.183·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.833i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.551 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.551 + 0.833i$
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: \((2,\ 1840,\ (\ :1),\ 0.551 + 0.833i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.081389264\)
\(L(\frac12)\) \(\approx\) \(2.081389264\)
\(L(2)\) 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 \( 1 + (-19.1 + 12.6i)T \)
good3 \( 1 - 1.73T + 9T^{2} \)
7 \( 1 + 1.02iT - 49T^{2} \)
11 \( 1 - 18.8iT - 121T^{2} \)
13 \( 1 - 0.710T + 169T^{2} \)
17 \( 1 + 6.02iT - 289T^{2} \)
19 \( 1 + 22.2iT - 361T^{2} \)
29 \( 1 - 22.1T + 841T^{2} \)
31 \( 1 + 5.70T + 961T^{2} \)
37 \( 1 + 27.1iT - 1.36e3T^{2} \)
41 \( 1 + 32.2T + 1.68e3T^{2} \)
43 \( 1 + 43.1iT - 1.84e3T^{2} \)
47 \( 1 - 77.3T + 2.20e3T^{2} \)
53 \( 1 - 24.6iT - 2.80e3T^{2} \)
59 \( 1 - 105.T + 3.48e3T^{2} \)
61 \( 1 - 16.7iT - 3.72e3T^{2} \)
67 \( 1 + 121. iT - 4.48e3T^{2} \)
71 \( 1 - 9.76T + 5.04e3T^{2} \)
73 \( 1 - 35.7T + 5.32e3T^{2} \)
79 \( 1 + 8.48iT - 6.24e3T^{2} \)
83 \( 1 + 37.9iT - 6.88e3T^{2} \)
89 \( 1 - 34.3iT - 7.92e3T^{2} \)
97 \( 1 - 108. iT - 9.40e3T^{2} \)
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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

−8.995864053377779529376049148420, −8.295857052013007435757409482459, −7.27534762897642932452183796264, −6.87104287365876846004837717991, −5.54798448590120284508120510593, −4.80643021156050638510885360896, −4.01547324911506281521141980503, −2.78217967406402453538757415084, −2.06194730133297285418968429579, −0.57408438836737860012480473768, 1.02079053922799042456933280804, 2.46532616584547745377612567681, 3.25114704267210036318669777279, 3.87669164442600557610705158662, 5.39532864428830306749181947747, 5.91024985900787675743473282038, 6.78124778397084633646920059164, 7.86516610629164422120234311335, 8.450554030423630737890677723925, 8.940671249528898435438199633785

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