Properties

Label 2-1840-5.4-c1-0-48
Degree $2$
Conductor $1840$
Sign $0.606 + 0.794i$
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  + 1.73i·3-s + (1.77 − 1.35i)5-s − 3.32i·7-s − 0.0262·9-s − 5.77·11-s − 1.10i·13-s + (2.36 + 3.09i)15-s + 0.893i·17-s + 2.42·19-s + 5.77·21-s + i·23-s + (1.31 − 4.82i)25-s + 5.17i·27-s − 4.11·29-s + 9.54·31-s + ⋯
L(s)  = 1  + 1.00i·3-s + (0.794 − 0.606i)5-s − 1.25i·7-s − 0.00874·9-s − 1.74·11-s − 0.305i·13-s + (0.609 + 0.798i)15-s + 0.216i·17-s + 0.557·19-s + 1.26·21-s + 0.208i·23-s + (0.263 − 0.964i)25-s + 0.995i·27-s − 0.763·29-s + 1.71·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.606 + 0.794i$
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: \((2,\ 1840,\ (\ :1/2),\ 0.606 + 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.663703352\)
\(L(\frac12)\) \(\approx\) \(1.663703352\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.77 + 1.35i)T \)
23 \( 1 - iT \)
good3 \( 1 - 1.73iT - 3T^{2} \)
7 \( 1 + 3.32iT - 7T^{2} \)
11 \( 1 + 5.77T + 11T^{2} \)
13 \( 1 + 1.10iT - 13T^{2} \)
17 \( 1 - 0.893iT - 17T^{2} \)
19 \( 1 - 2.42T + 19T^{2} \)
29 \( 1 + 4.11T + 29T^{2} \)
31 \( 1 - 9.54T + 31T^{2} \)
37 \( 1 + 7.69iT - 37T^{2} \)
41 \( 1 - 0.00418T + 41T^{2} \)
43 \( 1 + 9.97iT - 43T^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 + 6.25iT - 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 + 10.9iT - 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + 1.89iT - 73T^{2} \)
79 \( 1 + 0.216T + 79T^{2} \)
83 \( 1 + 5.38iT - 83T^{2} \)
89 \( 1 + 6.00T + 89T^{2} \)
97 \( 1 - 2.08iT - 97T^{2} \)
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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

−9.413878274918042554301140653044, −8.372218697889308812819392499498, −7.66600764078904222135991424803, −6.80997854947556887014755154929, −5.51199741000344436789593700853, −5.12191010301714677020391948679, −4.24453503414736375122268429729, −3.38339236175052042735170962813, −2.11450080795921441732554050191, −0.61633565659423295991926050197, 1.41072135574155377755105233294, 2.61969864924976145340974369367, 2.75791025198023463508048543064, 4.69786240238372450098900259202, 5.56375370519522376794678580970, 6.18577910550738521288206266715, 6.94154221261087124464335045623, 7.82259911577162600428616803572, 8.343582110378367154151212665189, 9.515628176570869802071674575506

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