↑

Properties

Label 2-1805-5.4-c1-0-114
Degree $2$
Conductor $1805$
Sign $0.809 - 0.586i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.96i·2-s − 0.187i·3-s − 1.84·4-s + (1.81 − 1.31i)5-s + 0.368·6-s − 0.677i·7-s + 0.297i·8-s + 2.96·9-s + (2.57 + 3.55i)10-s + 2.84·11-s + 0.346i·12-s − 4.76i·13-s + 1.32·14-s + (−0.246 − 0.339i)15-s − 4.27·16-s − 5.18i·17-s + ⋯
L(s)  = 1  + 1.38i·2-s − 0.108i·3-s − 0.924·4-s + (0.809 − 0.586i)5-s + 0.150·6-s − 0.255i·7-s + 0.105i·8-s + 0.988·9-s + (0.813 + 1.12i)10-s + 0.856·11-s + 0.100i·12-s − 1.32i·13-s + 0.354·14-s + (−0.0635 − 0.0877i)15-s − 1.06·16-s − 1.25i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.809 - 0.586i$
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: \((2,\ 1805,\ (\ :1/2),\ 0.809 - 0.586i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.234883841\)
\(L(\frac12)\) \(\approx\) \(2.234883841\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.81 + 1.31i)T \)
19 \( 1 \)
good2 \( 1 - 1.96iT - 2T^{2} \)
3 \( 1 + 0.187iT - 3T^{2} \)
7 \( 1 + 0.677iT - 7T^{2} \)
11 \( 1 - 2.84T + 11T^{2} \)
13 \( 1 + 4.76iT - 13T^{2} \)
17 \( 1 + 5.18iT - 17T^{2} \)
23 \( 1 - 1.05iT - 23T^{2} \)
29 \( 1 + 1.42T + 29T^{2} \)
31 \( 1 + 0.271T + 31T^{2} \)
37 \( 1 - 0.603iT - 37T^{2} \)
41 \( 1 + 6.73T + 41T^{2} \)
43 \( 1 + 5.62iT - 43T^{2} \)
47 \( 1 + 7.89iT - 47T^{2} \)
53 \( 1 + 6.88iT - 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 + 7.47T + 61T^{2} \)
67 \( 1 - 4.11iT - 67T^{2} \)
71 \( 1 - 7.60T + 71T^{2} \)
73 \( 1 - 16.1iT - 73T^{2} \)
79 \( 1 - 14.8T + 79T^{2} \)
83 \( 1 - 14.1iT - 83T^{2} \)
89 \( 1 - 8.23T + 89T^{2} \)
97 \( 1 + 5.72iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.185152728696523626913621703722, −8.427666822233032077846887668306, −7.59022774323862034788523236694, −6.95079566548874523469051055664, −6.28937902329613990964189895768, −5.32523291558040239771716539465, −4.91535267847769225866058237204, −3.73394362568609448998316362491, −2.22158252671573675258066013456, −0.888186004687412272469092456857, 1.54111475606042348713854060264, 1.87565068050735610951931432645, 3.12764370253257213762797344898, 4.00344094538568678453576472601, 4.68276009846982040577355440474, 6.21478393328050637752220416725, 6.57277829504935208573337574887, 7.57918455599975427887514912497, 9.093510235857574650885102670435, 9.257328705057087483642936046485

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