Properties

Label 2-1840-5.4-c1-0-39
Degree $2$
Conductor $1840$
Sign $i$
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  − 1.61i·3-s − 2.23·5-s − 1.85i·7-s + 0.381·9-s + 5.61·11-s + 2.61i·13-s + 3.61i·15-s − 0.854i·17-s − 0.145·19-s − 3·21-s i·23-s + 5.00·25-s − 5.47i·27-s + 9.70·29-s + 2.14·31-s + ⋯
L(s)  = 1  − 0.934i·3-s − 0.999·5-s − 0.700i·7-s + 0.127·9-s + 1.69·11-s + 0.726i·13-s + 0.934i·15-s − 0.207i·17-s − 0.0334·19-s − 0.654·21-s − 0.208i·23-s + 1.00·25-s − 1.05i·27-s + 1.80·29-s + 0.385·31-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.642900502\)
\(L(\frac12)\) \(\approx\) \(1.642900502\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23T \)
23 \( 1 + iT \)
good3 \( 1 + 1.61iT - 3T^{2} \)
7 \( 1 + 1.85iT - 7T^{2} \)
11 \( 1 - 5.61T + 11T^{2} \)
13 \( 1 - 2.61iT - 13T^{2} \)
17 \( 1 + 0.854iT - 17T^{2} \)
19 \( 1 + 0.145T + 19T^{2} \)
29 \( 1 - 9.70T + 29T^{2} \)
31 \( 1 - 2.14T + 31T^{2} \)
37 \( 1 - 9.70iT - 37T^{2} \)
41 \( 1 + 5.61T + 41T^{2} \)
43 \( 1 + 11.2iT - 43T^{2} \)
47 \( 1 - 1.70iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 2.85T + 61T^{2} \)
67 \( 1 - 5.23iT - 67T^{2} \)
71 \( 1 + 0.381T + 71T^{2} \)
73 \( 1 + 16.4iT - 73T^{2} \)
79 \( 1 + 7.70T + 79T^{2} \)
83 \( 1 + 7.70iT - 83T^{2} \)
89 \( 1 + 3.70T + 89T^{2} \)
97 \( 1 + 13.0iT - 97T^{2} \)
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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

−8.784385167738227097384581803497, −8.263551319419712130131247976950, −7.22715723877195782403349635457, −6.86019507879760787050634968953, −6.29720852306682266490098189385, −4.62806068679460302769009863625, −4.16650434119763488993331408300, −3.15955995242963240899943542471, −1.64866083417210551987370985058, −0.77518004565332744831613777496, 1.16496266223136356532337259221, 2.85545714268729118782634492363, 3.78083280748645323801804237318, 4.34425528069808022455527823395, 5.21023822604839441202975886528, 6.28413194147013180876171839497, 7.04226297846192532581681093713, 8.069319569926885834171896976353, 8.756681961364310166018026628000, 9.383226866345240309294557595151

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