Properties

Label 2-1045-209.208-c1-0-9
Degree $2$
Conductor $1045$
Sign $0.600 - 0.799i$
Analytic cond. $8.34436$
Root an. cond. $2.88866$
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  − 1.42·2-s − 1.05i·3-s + 0.0329·4-s − 5-s + 1.50i·6-s − 0.901i·7-s + 2.80·8-s + 1.89·9-s + 1.42·10-s + (2.88 + 1.63i)11-s − 0.0347i·12-s − 5.62·13-s + 1.28i·14-s + 1.05i·15-s − 4.06·16-s + 2.98i·17-s + ⋯
L(s)  = 1  − 1.00·2-s − 0.607i·3-s + 0.0164·4-s − 0.447·5-s + 0.612i·6-s − 0.340i·7-s + 0.991·8-s + 0.630·9-s + 0.450·10-s + (0.870 + 0.491i)11-s − 0.0100i·12-s − 1.56·13-s + 0.343i·14-s + 0.271i·15-s − 1.01·16-s + 0.723i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.600 - 0.799i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ 0.600 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5671800045\)
\(L(\frac12)\) \(\approx\) \(0.5671800045\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
11 \( 1 + (-2.88 - 1.63i)T \)
19 \( 1 + (0.563 - 4.32i)T \)
good2 \( 1 + 1.42T + 2T^{2} \)
3 \( 1 + 1.05iT - 3T^{2} \)
7 \( 1 + 0.901iT - 7T^{2} \)
13 \( 1 + 5.62T + 13T^{2} \)
17 \( 1 - 2.98iT - 17T^{2} \)
23 \( 1 + 5.84T + 23T^{2} \)
29 \( 1 - 2.10T + 29T^{2} \)
31 \( 1 + 9.55iT - 31T^{2} \)
37 \( 1 - 8.43iT - 37T^{2} \)
41 \( 1 + 4.89T + 41T^{2} \)
43 \( 1 - 10.7iT - 43T^{2} \)
47 \( 1 - 6.27T + 47T^{2} \)
53 \( 1 - 0.813iT - 53T^{2} \)
59 \( 1 - 13.9iT - 59T^{2} \)
61 \( 1 - 11.3iT - 61T^{2} \)
67 \( 1 - 5.05iT - 67T^{2} \)
71 \( 1 + 2.36iT - 71T^{2} \)
73 \( 1 + 4.24iT - 73T^{2} \)
79 \( 1 - 5.64T + 79T^{2} \)
83 \( 1 + 14.6iT - 83T^{2} \)
89 \( 1 - 13.7iT - 89T^{2} \)
97 \( 1 - 9.17iT - 97T^{2} \)
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

−10.05488246321027024536501164466, −9.305954922423489136253148808225, −8.188389003694687012086363198314, −7.67317639888235693655061627034, −7.08581012693906275717554830189, −6.09987037403362513859959717712, −4.50870120209882007431243721404, −4.06863928816270730132392144059, −2.16659891283780398214714196145, −1.14092035516051446393565564454, 0.42951466969411434919450461036, 2.08412481958388489615273784845, 3.60364071695858746694438347688, 4.56831105287606216356668339129, 5.22800395852759597953601797089, 6.88105042784076335389124179324, 7.31982528855303492543935558118, 8.417185680394192715580074802485, 9.097854952480542342776394321833, 9.660940482160843757498527323268

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