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Properties

Label 2-1840-460.459-c1-0-41
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  − 2.23·5-s + 4.79i·7-s − 3·9-s − 6.70·17-s − 4.79i·23-s + 5.00·25-s − 29-s − 10.7i·31-s − 10.7i·35-s + 11.1·37-s + 7·41-s + 9.59i·43-s + 6.70·45-s − 15.9·49-s + 2.23·53-s + ⋯
L(s)  = 1  − 0.999·5-s + 1.81i·7-s − 9-s − 1.62·17-s − 0.999i·23-s + 1.00·25-s − 0.185·29-s − 1.92i·31-s − 1.81i·35-s + 1.83·37-s + 1.09·41-s + 1.46i·43-s + 0.999·45-s − 2.28·49-s + 0.307·53-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: $\chi_{1840} (1839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4239936420\)
\(L(\frac12)\) \(\approx\) \(0.4239936420\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.79iT \)
good3 \( 1 + 3T^{2} \)
7 \( 1 - 4.79iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 6.70T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + 10.7iT - 31T^{2} \)
37 \( 1 - 11.1T + 37T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 - 9.59iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 2.23T + 53T^{2} \)
59 \( 1 + 10.7iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 4.79iT - 67T^{2} \)
71 \( 1 + 10.7iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 14.3iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 4.47T + 97T^{2} \)
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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

−9.016640925291164738914459390502, −8.287535745841263127007200216863, −7.77636886992075667481201846421, −6.39579841569836241510412957599, −6.01355556371422401659337887503, −4.92102137460758926290670495276, −4.15520192472543363413458049992, −2.79627492392428518324473270068, −2.35757200429384525315937316916, −0.18383230437144955010962535026, 1.02157955402819838808504326641, 2.72573984853211976124527586517, 3.78972420525058776239612961414, 4.28052466665405014407299457043, 5.26697566067275251932336283458, 6.52961426973584903350051045367, 7.14438571204482809724846478623, 7.79840409452487364724136024608, 8.580541436613405463971762996428, 9.329406917144981170938763396763

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