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Properties

Label 2-1840-5.4-c1-0-33
Degree $2$
Conductor $1840$
Sign $0.732 - 0.680i$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.486i·3-s + (1.52 + 1.63i)5-s − 1.80i·7-s + 2.76·9-s + 2.90·11-s + 2.25i·13-s + (−0.796 + 0.740i)15-s + 2.14i·17-s + 0.339·19-s + 0.877·21-s − i·23-s + (−0.369 + 4.98i)25-s + 2.80i·27-s + 5.60·29-s − 5.92·31-s + ⋯
L(s)  = 1  + 0.280i·3-s + (0.680 + 0.732i)5-s − 0.682i·7-s + 0.921·9-s + 0.876·11-s + 0.624i·13-s + (−0.205 + 0.191i)15-s + 0.520i·17-s + 0.0779·19-s + 0.191·21-s − 0.208i·23-s + (−0.0738 + 0.997i)25-s + 0.539i·27-s + 1.04·29-s − 1.06·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.732 - 0.680i$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ 0.732 - 0.680i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.285707829\)
\(L(\frac12)\) \(\approx\) \(2.285707829\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 - 1.63i)T \)
23 \( 1 + iT \)
good3 \( 1 - 0.486iT - 3T^{2} \)
7 \( 1 + 1.80iT - 7T^{2} \)
11 \( 1 - 2.90T + 11T^{2} \)
13 \( 1 - 2.25iT - 13T^{2} \)
17 \( 1 - 2.14iT - 17T^{2} \)
19 \( 1 - 0.339T + 19T^{2} \)
29 \( 1 - 5.60T + 29T^{2} \)
31 \( 1 + 5.92T + 31T^{2} \)
37 \( 1 + 8.98iT - 37T^{2} \)
41 \( 1 - 1.89T + 41T^{2} \)
43 \( 1 + 9.47iT - 43T^{2} \)
47 \( 1 - 7.83iT - 47T^{2} \)
53 \( 1 + 6.47iT - 53T^{2} \)
59 \( 1 - 5.17T + 59T^{2} \)
61 \( 1 + 9.12T + 61T^{2} \)
67 \( 1 - 9.25iT - 67T^{2} \)
71 \( 1 - 4.60T + 71T^{2} \)
73 \( 1 - 11.3iT - 73T^{2} \)
79 \( 1 - 7.94T + 79T^{2} \)
83 \( 1 - 5.37iT - 83T^{2} \)
89 \( 1 + 12.9T + 89T^{2} \)
97 \( 1 + 2.43iT - 97T^{2} \)
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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

−9.437597918278792185510983873927, −8.788256472294338794364487162415, −7.50270821243286141156621914116, −6.95102287931364286047536938758, −6.32862937803217331348438962958, −5.32153399235773970066522820414, −4.14266789144065730274494246599, −3.71832020933038248244593690352, −2.29828139215640907898175892403, −1.28099793020103856137944449013, 1.01610558145078036519581777756, 1.92488127742330185084175562371, 3.10166943391414579182667229150, 4.37333664262803874722193381546, 5.08864317109116551299693888052, 6.00032438350330704934114682254, 6.66009557779807081552801788950, 7.62608473049801934284254487851, 8.469047250271380002399253399457, 9.216201593030520180477515600134

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