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Properties

Label 2-1805-5.4-c1-0-91
Degree $2$
Conductor $1805$
Sign $0.995 - 0.0983i$
Analytic cond. $14.4129$
Root an. cond. $3.79644$
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  − 0.805i·2-s + 1.95i·3-s + 1.35·4-s + (2.22 − 0.219i)5-s + 1.57·6-s − 1.03i·7-s − 2.69i·8-s − 0.835·9-s + (−0.177 − 1.79i)10-s + 1.80·11-s + 2.64i·12-s − 2.36i·13-s − 0.835·14-s + (0.430 + 4.35i)15-s + 0.529·16-s + 6.49i·17-s + ⋯
L(s)  = 1  − 0.569i·2-s + 1.13i·3-s + 0.675·4-s + (0.995 − 0.0983i)5-s + 0.643·6-s − 0.391i·7-s − 0.954i·8-s − 0.278·9-s + (−0.0559 − 0.566i)10-s + 0.544·11-s + 0.764i·12-s − 0.655i·13-s − 0.223·14-s + (0.111 + 1.12i)15-s + 0.132·16-s + 1.57i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.995 - 0.0983i$
Analytic conductor: \(14.4129\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ 0.995 - 0.0983i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.783988519\)
\(L(\frac12)\) \(\approx\) \(2.783988519\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.22 + 0.219i)T \)
19 \( 1 \)
good2 \( 1 + 0.805iT - 2T^{2} \)
3 \( 1 - 1.95iT - 3T^{2} \)
7 \( 1 + 1.03iT - 7T^{2} \)
11 \( 1 - 1.80T + 11T^{2} \)
13 \( 1 + 2.36iT - 13T^{2} \)
17 \( 1 - 6.49iT - 17T^{2} \)
23 \( 1 - 7.26iT - 23T^{2} \)
29 \( 1 + 2.22T + 29T^{2} \)
31 \( 1 + 5.25T + 31T^{2} \)
37 \( 1 + 6.67iT - 37T^{2} \)
41 \( 1 - 4.43T + 41T^{2} \)
43 \( 1 + 6.26iT - 43T^{2} \)
47 \( 1 + 8.38iT - 47T^{2} \)
53 \( 1 - 0.0601iT - 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 - 6.49iT - 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + 8.23iT - 73T^{2} \)
79 \( 1 - 6.11T + 79T^{2} \)
83 \( 1 + 4.95iT - 83T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 - 18.4iT - 97T^{2} \)
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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

−9.487119058804351973053315912775, −8.928835634051338986121965779114, −7.66810996030872131452022209301, −6.84728242599630610894228259095, −5.85207626739690426054038169404, −5.31697375247047026133635615715, −3.83105086668779172520547748870, −3.67372431602947322995530911166, −2.22919534993742406097048594251, −1.32126691646878489923596407779, 1.22834855028384112158023005930, 2.20491231914552228705720756657, 2.80178150796030674044126086427, 4.59074732506340216481718097209, 5.58056284651584303367517855197, 6.35130203563316990145625671965, 6.79820601575862652765332524878, 7.37780044853372591777540292226, 8.306749457864923473997539646054, 9.157149978279427883857898543147

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